Solving Absolute Value Equations and Inequalities

1.7
Solving Absolute Value
Equations and Inequalities
Goals p Solve absolute value equations and inequalities.
p Use absolute values in real-life problems.
Your Notes
VOCABULARY
Absolute value of a real number The distance the
number is from 0 on a number line
SOLVING AN ABSOLUTE VALUE EQUATION
The absolute value equation ⏐ax b⏐ c, where c > 0, is
equivalent to the compound statement ax b c
or ax b c .
Example 1
Solving an Absolute Value Equation
⏐4x 2⏐ 6
Original equation
4x 2 6 or 4x 2 6 Expression can be 6
or 6 .
4x 4 or 4x 8 Subtract 2 from each
side.
x 1 or x 2 Divide each side by 4 .
TRANSFORMATIONS OF ABSOLUTE VALUE INEQUALITIES
p The inequality ⏐ax b⏐< c, where c > 0, means that
ax b is between c and c. This is equivalent to
c < ax b < c .
p The inequality ⏐ax b⏐> c, where c > 0, means that
ax b is beyond c and c. This is equivalent to
ax b < c or ax b > c .
In the first transformation, < can be replaced by ≤ . In the
second transformation, > can be replaced by ≥ .
22 Algebra 2 Notetaking Guide • Chapter 1

Your Notes
Example 2
Solving an Inequality of the form⏐ax b⏐ < c
Solve ⏐2x 3⏐ < 3.
Solution
⏐2x 3⏐ < 3 Write original inequality.
3 < 2x 3 < 3 Write equivalent compound inequality.
0 < 2x < 6 Add 3 to each expression.
0 < x < 3 Divide each expression by 2 .
The solutions are all real numbers greater than 0 and less
than 3 . Check several solutions in the original inequality.
Graph the solution below.
2 1
0 1 2 3 4 5
Example 3
Solving an Inequality of the form ⏐ax b⏐ ≥ c
Solve ⏐2x 7⏐ ≥ 3.
Solution
This absolute value inequality is equivalent to 2x 7 ≤ 3
or 2x 7 ≥ 3 .
Solve first inequality
Solve second inequality
2x 7 ≤ 3 Write inequality. 2x 7 ≥ 3
2x ≤ 10 Subtract 7 2x ≥ 4
from each side.
x ≤ 5 Divide each side x ≥ 2
by 2 .
The solutions are all real numbers less than or equal to 5
or greater than or equal to 2 . Check several solutions in
the original inequality. Graph the solution below.
7
6 5
4 3 2 1
0
Lesson 1.7 • Algebra 2 Notetaking Guide 23

Your Notes
Checkpoint Solve the equation or inequality.
1.⏐5x 2⏐ 3 2.⏐3x 4⏐ < 2 3.⏐2x 3⏐ ≥ 5
1 5 ,1 2 < x < 2 x ≤ 1 or
3
x ≥ 4
Example 4
Write a Model for Tolerance
A dog food manufacturer has a tolerance of 0.25 pound per
bag of dog food advertised as weighing 5 pounds. Write and
solve an absolute value inequality that describes the
acceptable weights for “5 pound” bags.
Verbal
Model
Actual weight
Ideal weight
≤
Tolerance
Labels
Actual weight x
(pounds)
Ideal weight 5
(pounds)
Tolerance 0.25
(pounds)
Algebraic
Model
⏐x 5⏐ ≤ 0.25
Write algebraic model.
0.25 ≤ x 5 ≤ 0.25 Write equivalent
compound inequality.
4.75 ≤ x ≤ 5.25 Add 5 to each
expression.
The weights can range between 4.75 pounds and
5.25 pounds, inclusive.
Checkpoint Complete the following exercise.
Homework
4. A toy manufacturer has a tolerance of 0.1 inch on a ball
that is supposed to have a diameter of 1 inch. Write and
solve an inequality describing the acceptable diameters
for a ball.
⏐d 1⏐ ≤ 0.1, where d represents the actual
diameter; 0.9 ≤ d ≤ 1.1
24 Algebra 2 Notetaking Guide • Chapter 1