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562 (10–38) Chapter 10 Quadratic Equations,Functions,and Inequalities
4 2
GRAPHING CALCULATOR 67. x 116x 1600 0
EXERCISES
2 2 2
Solve each equation by locating the x-intercepts on the 68. (x 3x) 7(x 3x) 9 0
graph of a corresponding function. Round approximate an-
swers to two decimal places. 2 12
69. x 3x 120
66. (5x 7)2 (5x 7) 6 0
10.5 QUADRATIC AND RATIONAL
INEQUALITIES
Inthis In this section we solve inequalities involving quadratic polynomials. We use a new
technique based on the rules for multiplying real numbers.
section
● Solving Quadratic Solving Quadratic Inequalities with a Sign Graph
Inequalities with a Sign An inequality involving a quadratic polynomial is called a quadratic inequality.
Graph
● Solving Rational Inequalities
with a Sign Graph Quadratic Inequality
● Quadratic Inequalities That Aquadratic inequality is an inequality of the form
Cannot Be Factored
● Applications ax2 bx c 0,
where a, b, and c are real numbers with a
0. The inequality symbols , ,
and may also be used.
If we can factor a quadratic inequality, then the inequality can be solved with a sign
graph, which shows where each factor is positive, negative, or zero.
EXAMPLE 1 Solving a quadratic inequality
2
Use a sign graph to solve the inequality x 3x 10 0.
Solution
Because the left-hand side can be factored, we can write the inequality as
(x 5)(x 2) 0.
This inequality says that the product of x 5 and x 2 is positive. If both factors
are negative or both are positive, the product is positive. To analyze the signs of
each factor, we make a sign graph as follows. First consider the possible values of
the factor x 5:
Value Where On the number line
x 5 0 if x 5 Put a 0 above 5.
x 5 0ifx5 Put signs to the right of 5.
x 5 0ifx 5 Put signs to the left of 5.
10.5 Quadratic and Rational Inequalities (10–39) 563
calculator The sign graph shown in Fig. 10.9 for the factor x 5 is made from the informa-
tion in the preceding table.
close-up (x + 5) negative here (x + 5) positive here
x + 5 – ––––––––––––0 +++++++++++++
UseYtoset y x 5and
1
y x2.Nowmakeatable – – – – – – – – – – –
2 11 10 9 8 7 615 4 3 2 10
and scroll through the table. FIGURE 10.9
Thetablenumericallysupports
thesigngraphinFig.10.10. Now consider the possible values of the factor x 2:
Value Where On the number line
x 2 0 if x 2 Put a 0 above 2.
x 2 0ifx2 Put signs to the right of 2.
Note that the graph of x 2 0ifx 2 Put signs to the left of 2.
2
y x 3x 10 is above
the x-axis when x 5or We put the information for the factor x 2 on the sign graph for the factor x 5
whenx 2. as shown in Fig. 10.10. We can see from Fig. 10.10 that the product is positive if
10 x 5and the product is positive if x 2. The solution set for the quadratic
inequality is shown in Fig. 10.11. Note that 5 and 2 are not included in the graph
because for those values of x the product is zero. The solution set is (
, 5)
–84
(2,
).
–15 Positive product because Positive product because
both factors are negative both factors are positive
x – 2 – ––––––––––––––––––––––0 +++++++++
x + 5 – ––––––––0 +++++++++++++++++++++++
– – – – – – – – –
9 8 7 615 4 3 2 10 23456
FIGURE 10.10
– – – – – – – – –
9 8 7 6154 3 2 10 23456
FIGURE 10.11 ■
In the next example we will make the procedure from Example 1 a bit more
efficient.
EXAMPLE 2 Solving a quadratic inequality
2
Solve 2x 5x 3 and graph the solution set.
Solution
Rewrite the inequality with 0 on one side:
2
2x 5x3 0
(2x 1)(x 3) 0 Factor.
564 (10–40) Chapter 10 Quadratic Equations,Functions,and Inequalities
calculator Examine the signs of each factor:
2x 1 0 if x 1
2
close-up 2x 1 0 if x 1
2
Use Y to set y 2x 1
1 1
and y x 3. The table of 2x 1 0 if x
2 2
values for y and y supports
1 2
the sign graph in Fig. 10.12. x 3 0 if x 3
x 3 0 if x 3
x 3 0 if x 3
Make a sign graph as shown in Fig. 10.12. The product of the factors is negative be-
1
tween 3 and , when one factor is negative and the other is positive. The product
2 1 1
is 0 at 3 and at . So the solution set is the interval 3, . The graph of the so-
2 2
Note that the graph of lution set is shown in Fig. 10.13.
y 2x2 5x 3 is below
the x-axis when x is between x + 3 – ––––––––0 +++++++++++++++
– ––––––––––––––– ++++++++
1 2x – 1 0
3 and .
2 1
– – – – – – – —
7 615 4 3 2 10 234
10 2
Positive product Negative product Positive product
–62 FIGURE 10.12
1
—
2
–10
– – – – – – –
7 6154 3 2 10 234
FIGURE 10.13 ■
We summarize the strategy used for solving a quadratic inequality as follows.
Strategy for Solving a Quadratic Inequality
with a Sign Graph
1. Write the inequality with 0 on the right.
2. Factor the quadratic polynomial on the left.
3. Make a sign graph showing where each factor is positive, negative, or zero.
4. Use the rules for multiplying signed numbers to determine which regions
satisfy the original inequality.
Solving Rational Inequalities with a Sign Graph
The inequalities
x 2 2, 2x 3 0and 2 1
x 3 x 5 x 4 x 1
are called rational inequalities. When we solve equations that involve rational ex-
pressions, we usually multiply each side by the LCD. However, if we multiply each
side of any inequality by a negative number, we must reverse the inequality, and
10.5 Quadratic and Rational Inequalities (10–41) 565
when we multiply by a positive number, we do not reverse the inequality. For this
reason we generally do not multiply inequalities by expressions involving variables.
The values of the expressions might be positive or negative. The next two examples
show how to use a sign graph to solve rational inequalities that have variables in the
denominator.
EXAMPLE 3 Solving a rational inequality
x 2
Solve 2 and graph the solution set.
x 3
Solution
helpful hint We do notmultiply each side by x 3. Instead, subtract 2 from each side to get 0
on the right:
By getting 0 on one side of the x 2
inequality, we can use the 2 0
rules for dividing signed num- x 3
bers.The only way to obtain a x 2 2(x 3) 0 Get a common denominator.
negative result is to divide
x 3 x 3
numbers with opposite signs. x 2 2x 6
0 Simplify.
x 3 x 3
x 2 2x 6 0 Subtract the rational expressions.
x 3
x8 0 The quotient of x8 and x3 is less
x 3 than or equal to 0.
calculator Examine the signs of the numerator and denominator:
x 3 0 if x 3 x80 if x8
close-up x 3 0 if x 3 x80 if x 8
x8 x 3 0 if x 3 x8 0 if x8
Graph y = to support
x 3 MakeasigngraphasshowninFig.10.14.Usingtherulefordividing signed num-
the conclusion that y 0 bers and the sign graph, we can identify where the quotient is negative or zero. The
when x 3 or x 8. solution set is (
,3) [8,
). Note that 3 is not in the solution set because the
5 quotient is undefined if x 3. The graph of the solution set is shown in Fig. 10.15.
–x + 8 +++++++++++++++++ 0 – ––– –––
–3 12 x – 3 – ––––––0 + ++++++++++++++++
01234567891011
–5 Negative quotient Negative quotient
FIGURE 10.14
01234567891011
FIGURE 10.15 ■
CAUTION Remember to reverse the inequality sign when multiplying or
dividing by a negative number. For example, x 3 0 is equivalent to x 3. But
x80is equivalent to x 8, or x 8.
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