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Section 2.6 Solving Inequalities Algebraically and Graphically 59
Course Number
Section 2.6 Solving Inequalities Algebraically and
Graphically Instructor
Objective: In this lesson you learned how to solve linear inequalities, Date
inequalities involving absolute values, polynomial
inequalities, and rational inequalities.
Important Vocabulary Define each term or concept.
Solutions of an inequality sss ssssss ss sss ssssssss sss sssss sss ssssssssss ss sssss
Graph of an inequality sss sss ss sss ssssss ss sss ssss ssssss ssss ssss sssssssss sss
ssssssss sss ss ss sssssssssss
Double inequality ss ssssssssss ssss ssssssssss sss sssssssssssss
Critical numbers sss ssssssss ssss ssss sss ssssssssss ss s ssssssssss ssssssssss sssss ss
sssss
Test intervals ssss sssssssss sssss sss ssss ssssss ssss ss sssss sss ssssssssss sss ss ssss
ssssssss
I. Properties of Inequalities (Pages 219−220) What you should learn
How to recognize
Solving an inequality in the variable x means . . . sssssss sss sss properties of inequalities
ssssss ss s sss sssss sss ssssssssss ss sssss
Numbers that are solutions of an inequality are said to
sssssss s the inequality.
To solve a linear inequality in one variable, use the s sssssssss
ss ssssssssssss to isolate the variable.
When each side of an inequality is multiplied or divided by a
negative number, . . . sss sssssssss ss sss ssssssssss ssssss ssss
ss ssssssss ss sssss ss ssssssss s ssss ssssssssss
Two inequalities that have the same solution set are
ssssssssss ssssssssssss .
Complete the list of Properties of Inequalities given below.
1) Transitive Property: a < b and b < c → s s s s
2) Addition of Inequalities: a < b and c < d → s s s s s s s s
Larson/Hostetler/Edwards Algebra and Trigonometry: A Graphing Approach, Fifth Edition Student Success Organizer
Copyright © Houghton Mifflin Company. All rights reserved.
60 Chapter 2 Solving Equations and Inequalities
3) Addition of a Constant c: a < b → s s s s s s s s
4) Multiplication by a Constant c:
For c > 0, a < b → ss s ss s
For c < 0, a < b → ss s ss s
II. Solving a Linear Inequality (Pages 220−221) What you should learn
How to use properties of
Describe the steps that would be necessary to solve the linear inequalities to solve
inequality 7x − 2<9x +8. linear inequalities
sss s ss ssss sssss ssssssss ss ssss ssss sssss sss sssssss ssss ssssss
ssssss ssss ssss ss s s sss sssssss sss sssssssssss sssss sss ssssssss
sss ss ss sssssssss
To use a graphing utility to solve the linear inequality
7x−2<9x+8, . . . sssss ss s ss s s sss ss s ss s s ss sss ssss
sssssss sssssss sss sss sssssssss sssssss ss sss ssssssss sssssss ss
ssss sss sssss ss sssssssssssss ssssssss sssss sss sssss ss ss ssss
sssss sss sssss ss sss sssss sss ssssssss ssss
The two inequalities − 10 < 3x and 14 ≥ 3x can be rewritten as
the double inequality s ss s ss s ss .
III. Inequalities Involving Absolute Value (Page 222) What you should learn
How to solve inequalities
Let x be a variable or an algebraic expression and let a be a real involving absolute values
number such that a ≥ 0. The solutions of x < a are all values of
x that sss sssssss s s sss s . The solutions of
x >a are all values of x that sss ssss ssss s s ss sssssss
ssss s .
Example 1: Solve the inequality: x +11 −4≤0
ss sss s ss
The symbol ∪ is called a sssss symbol and is used to
denote sss sssssssss ss sss ssss .
Larson/Hostetler/Edwards Algebra and Trigonometry: A Graphing Approach, Fifth Edition Student Success Organizer
Copyright © Houghton Mifflin Company. All rights reserved.
Section 2.6 Solving Inequalities Algebraically and Graphically 61
Example 2: Write the following solution set using interval
notation: x > 8 or x < 2
ss ss ss s sss ss
IV. Polynomial Inequalities (Pages 223−225) What you should learn
How to solve polynomial
Where can polynomials change signs? inequalities
ssss ss sss ssssss sss ssssssss ssss ssss sss ssssssssss sssss ss sssss
Between two consecutive zeros, a polynomial must be . . .
ssssssss ssssssss ss ssssssss sssssssss
When the real zeros of a polynomial are put in order, they divide
the real number line into . . . sssssssss ss sssss sss ssssssssss
sss ss ssss ssssssss
These zeros are the ssssssss sssssss of the inequality,
and the resulting open intervals are the ssss sssssssss
.
Complete the following steps for determining the intervals on
which the values of a polynomial are entirely negative or entirely
positive:
1) ssss sss ssss sssss ss sss sssssssssss sss sssssss sss sssss
ss ssssssssss ssssss sss sssss ss s ssssssssss sss sss
ssssssss ssssssss
2) sss sss ssssssss sssssss ss sssssssss sss ssss ssssssssss
3) ssssss sss ssssssssssssss sssssss ss ssss ssss ssssssss sss
ssssssss sss ssssssssss ss ssss ssssss ss sss sssss ss sss
ssssssssss ss sssssssss sss ssssssssss ssss ssss ssssssss
ssssss sss sssss sssssss ss sss sssssssss ss sss sssss ss sss
ssssssssss ss sssssssss sss ssssssssss ssss ssss ssssssss
ssssss sss sssss sssssss ss sss sssssssss
To approximate the solution of the polynomial inequality
3x2 + 2x −5<0 from a graph, . . . sssss sss ssssssssss
ssssssssss s s sss s ss s s sss ssssss sss sssssss ss sss sssss ssss ss
sssss sss sssssss
Larson/Hostetler/Edwards Algebra and Trigonometry: A Graphing Approach, Fifth Edition Student Success Organizer
Copyright © Houghton Mifflin Company. All rights reserved.
62 Chapter 2 Solving Equations and Inequalities
If a polynomial inequality is not given in general form, you
should begin the solution process by . . . sssssss sss
ssssssssss ss sssssss sssssssss sss ssssssssss ss sss ssss sss ssss ss
sss sssss sssss
y
5
Example 3: Solve x2 + x −20≥0.
sss ss s ss s sss ss 3
1
x
Example 4: Use a graph to solve the polynomial inequality -5 -3 -1 1 3 5
-1
−x2 −6x−9>0.
ss -3
-5
V. Rational Inequalities (Page 226) What you should learn
How to solve rational
To extend the concepts of critical numbers and test intervals to inequalities
rational inequalities, use the fact that the value of a rational
expression can change sign only at its sssss and its
sssssssss ssssss . These two types of numbers make
up the ssssssss sssssss of a rational inequality.
To solve a rational inequality, . . . sssss sssss sss ssssssss
ssssssssss ss ssssssss sssss ssss ssss sss sssss sss sssssssss ssssss
ss sss sssssssss ssssssss sssssssssss ssss sss sssssssssss ssss
sssssssss sss ssss s sssss ssss ssss ssssssss ss sss sssssssssss ssssss
sss ssss sssssssss ssss sssssss sss ssssssssss ss sss sssss sss
ssss
Example 5: Solve 3x+15 ≤0.
x − 2
sss ss ss
Homework Assignment
Page(s)
Exercises
Larson/Hostetler/Edwards Algebra and Trigonometry: A Graphing Approach, Fifth Edition Student Success Organizer
Copyright © Houghton Mifflin Company. All rights reserved.
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