Number Systems
                                                                and Inequalities
                                  Quadratic Inequalities
             Aim
             To demonstrate how to solve quadratic inequalities.
             Learning Outcomes
             At the end of this section you will be able to:
                • Distinguish between quadratic equations and inequalities,
                • Solve quadratic inequalities.
               Theinequalities that we have dealt with so far are called “linear” inequalities because
            they only deal with linear expressions like x − 2 > 0. An inequality where x2 is the
                                                                       2
            highest-degree term is called a quadratic inequality. In general, ax +bx+c = 0 is a
                                     2
            quadratic equation, while ax +bx+c ≤ 0 is an example of a quadratic inequality.
               Solving a quadratic inequality means that we are looking for the range of values of
            x where the inequality is greater than or less than zero. The first step is to factorise the
            quadratic inequality as if it was just a quadratic equation.
            Example 1
               Find the set of values of x for which 1 + 2x − 3x2 > 0.
               Factorise the quadratic,
                               1+2x−3x2>0 ⇒ 3x2−2x−1<0,
                                               ⇒ (3x+1)(x−1)<0.
               What this result tells us is that the product of two values when multiplied together
            is negative. This can only occur when one of the values is negative and the other one
            positive. There are two possibilities that need to be looked at:
               1. (3x+1) < 0 and (x−1) > 0, or
               2. (3x+1) > 0 and (x−1) < 0.
                                                 1
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                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           and Inequalities
                                                                                                                                                                                                                                                                                                                                                                                           Looking at case 1 we have
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            (3x+1)<0                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              and                                                                                                                                                                                                                                                                                   (x−1)>0,
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         ⇒x<−1                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    and                                                                                                                                                                                                                                                                                   x>1.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          3
                                                                                                                                                                                                                                                                                                                                                                                           and looking at case 2 reveals that
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            (3x+1)>0                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              and                                                                                                                                                                                                                                                                                   (x−1)<0,
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         ⇒x>−1                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    and                                                                                                                                                                                                                                                                                   x<1.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          3
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         It can be seen by a quick inspection that the results from case 1 do not make any
                                                                                                                                                                                                                                                                                                                                                                                           mathematical sense as there are no values of x that are less than −1 and at the same
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      3
                                                                                                                                                                                                                                                                                                                                                                                           time greater than 1. On the other hand, the conditions from case 2 are that x is greater
                                                                                                                                                                                                                                                                                                                                                                                           than −1 and at the same time less than 1 which is possible. Therefore the final answer
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     3
                                                                                                                                                                                                                                                                                                                                                                                           is
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     x>−1                                                                                                                                                                                                                                                                                                                                              and                                                                                                                                                                                                                               x<1.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    3
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         This tells us the value of x where the graph is below the x axis (y < 0).
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              y = 3x2 −2x−1
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              ✁✂✄                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      2                                                                                                                                                                                            ✁
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      Figure 1: The graph of y = 3x −2x−1
                                                                                                                                                                                                                                                                                                                                                                                           Example 2
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         Find the values of x for which 2x2 + 5x − 3 > 0.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           2
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        2x +5x−3>0⇒(2x−1)(x+3)>0.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         In this case the product of (2x − 1) and (x + 3) must be greater than zero so either
                                                                                                                                                                                                                                                                                                                                                                                           both of them must be positive or else both of them must be negative. Therefore
                                                                                                                                                                                                                                                                                                                                                                                                                                                                       1. (2x−1) > 0 and (x+3) > 0 or
                                                                                                                                                                                                                                                                                                                                                                                                                                                                       2. (2x−1) < 0 and (x+3) < 0.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   2
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      Number Systems
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           and Inequalities
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         Looking at case 1:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            (2x−1)>0                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               and                                                                                                                                                                                                                                                                                   (x+3)>0,
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           ⇒x>1                                                                                                                                                                                                                                                                                                                                                                                                                                    and                                                                                                                                                                                                                                                                                    x>−3.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           2
                                                                                                                                                                                                                                                                                                                                                                                           and looking at case 2 reveals that
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            (2x−1)<0                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               and                                                                                                                                                                                                                                                                                   (x+3)<0,
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           ⇒x<1                                                                                                                                                                                                                                                                                                                                                                                                                                    and                                                                                                                                                                                                                                                                                    x<−3.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           2
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         It is clear to see that in case 1 the only values of x which satisfy both conditions
                                                                                                                                                                                                                                                                                                                                                                                           occur when x > 1. Similarly for case 2, the only values of x which satisfy both of these
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      2
                                                                                                                                                                                                                                                                                                                                                                                           conditions is x < −3.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         Therefore the final answer is
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           x<−3                                                                                                                                                                                                                                                                                                                                  or                                                                                                                                                                                  x>1.
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   2
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         This tells us the value of x where the graph is above the x axis (y > 0).
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              y = 2x2 +5x−3
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                ✁                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         ✂
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        2                                                                                                                                                                                                     ✄☎
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       Figure 2: The graph of y = 2x +5x−3
                                                                                                                                                                                                                                                                                                                                                                                           Related Reading
                                                                                                                                                                                                                                                                                                                                                                                           Morris, O.D. 1987. Text & Tests 1. The Celtic Press.
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