Section 1.7
Solving Absolute Value Equations
and Inequalities
Definition of Absolute Value
Absolute value of a number x, written |x|, is the
distance the number is from 0 on a number line.
if x is positive
 x,

x  0,
if x  0
 x, if x is negative

Interpreting Absolute Value Equations
Equation:
|x| = |x – 0| = k
Meaning:
The distance between x and 0 is k.
k
Graph:
−k
k
0
k
Solutions:
x – 0 = −k
x = −k
or
or
x–0=k
x=k
Equation:
|x − b| = k
Meaning:
The distance between x and b is k.
k
Graph:
b−k
k
b
b+k
Solutions:
x – b = −k
or
x = b − k or
x–b=k
x=b+k
Example 1
Solve |x – 9| = 3. Graph the solutions.
x – 9 = -3
x=6
x–9=3
x = 12
or
or
3
2
3
4
5
6
7
3
8
9 10 11 12 13 14 15
Solving an Absolute Value Equation
Use these steps to solve an absolute value
equation |ax + b| = c where c > 0.
1. Write two equations:
ax + b = -c or ax + b = c
2. Solve each equation.
3. Check each solution in the original
absolute value equation.
Example 2
Solve |4x + 12| = 28.
4x + 12 = -28
or
4x = -40
or
x = -10
or
4x + 12 = 28
4x = 16
x=4
Extraneous Solutions
An extraneous solution is an apparent solution
that must be rejected because it does not satisfy
the original equation.
Example 3
Solve |4x + 10| = 6x.
4x + 10 = -6x
or
10x = -10
or
x = -1
or
4x + 10 = 6x
-2x = -10
x=5
|4(-1) + 10| = 6(-1)
6 = -6
|4(5) + 10| = 6(5)
30 = 30
Absolute Value Inequalities
1.
|ax + b| < c
Equivalent Form: -c < ax + b < c
Graph of Solution:
2.
|ax + b| ≤ c
Equivalent Form: -c ≤ ax + b ≤ c
Graph of Solution:
3.
|ax + b| > c
Equivalent Form: ax + b < -c or ax + b > c
Graph of Solution:
4.
|ax + b| ≤ c
Equivalent Form: ax + b ≤ -c or ax + b ≥ c
Graph of Solution:
Example 4
Solve |3x – 7| ≥ 5.
3x – 7 ≤ -5
3x ≤ 2
2
x
3
-1
0
1
2
The graph the solution.
or
3x – 7 ≥ 5
or
3x ≥ 12
x 4
or
3
4
5
6
7
Example 5
A food manufacturer specifies that every familysize box of cereal should have a net weight of 25
ounces, with a tolerance of 1.2 ounces. Write
and solve an absolute value inequality that
describes the acceptable net weights for the
cereal in a family-size box.
Tolerance is the maximum acceptable deviation
of an item from some ideal or mean
measurement.
Let w = the actual weight of a box of cereal
Verbal model:
|actual weight – ideal weight| ≤ tolerance
|w – 25| ≤ 1.2
-1.2 ≤ w – 25 ≤ 1.2
23.8 ≤ w ≤ 26.2
The net weight for a family-size box of cereal is
between 23.8 ounces and 26.2 ounces, inclusive.
Example 6
You have found that your new winter coat is
comfortable to wear when the outdoor
temperature is between 10°F and 42°F, inclusive.
Write an absolute value inequality for this
temperature range, where t represents the
temperature in degrees Fahrenheit.
1.
Calculate the means of the extreme
temperatures.
42  10
Mean of extremes 
 26
2
2. Find the tolerance.
tolerance = 42 – 26 = 16
3. Write a verbal model.
|Actual temperature – Mean temperature| ≤
tolerance
|t – 26| ≤ 16