1. Solving Absolute Value Equations
Today’s objectives:
1. I will solve absolute value
equations.
2. I will use absolute value equations
to solve real-life problems.

2. What is Absolute Value?
• The absolute value of a number is the
number of units it is from zero on the
number line. Since distance is positive,
absolute value is NEVER NEGATIVE.
• 5 and -5 have the same absolute value.
• The symbol |x| represents the absolute value
of the number x.

3. Absolute Value
• |-8| = 8
• |4| = 4
• You try:
• |15| = ?
• |-23| = ?
• Absolute Value can also be defined as :
• if a >0, then |a| = a
• if a < 0, then |-a| = a

4. We can evaluate expressions that contain
absolute value symbols.
• Think of the | | bars as grouping symbols.
• Evaluate |9x -3| + 5 if x = -2
|9(-2) -3| + 5
|-18 -3| + 5
|-21| + 5
21+ 5=26

5. Solving absolute value equations
• First, isolate the absolute value expression.
• Set up two equations to solve.
• For the first equation, drop the absolute value
bars and solve the equation.
• For the second equation, drop the bars, negate
the opposite side, and solve the equation.
• Always check the solutions.

6. Equations may also contain absolute
value expressions
• When solving an equation, isolate the absolute value
expression first.
• Rewrite the equation as two separate equations.
• Consider the equation | x | = 3. The equation has two solutions
since x can equal 3 or -3.
• Solve each equation.
• Always check your solutions.
Example: Solve |x + 8| = 3
x + 8 = 3 and x + 8 = -3
x = -5 x = -11
Check: |x + 8| = 3
|-5 + 8| = 3 |-11 + 8| = 3
|3| = 3 |-3| = 3
3=3 3=3

7. Now Try These
• Solve |y + 4| - 3 = 0
|y + 4| = 3 You must first isolate the variable by adding
3 to both sides.
• Write the two separate equations.
y+4=3 & y + 4 = -3
y = -1 y = -7
• Check: |y + 4| - 3 = 0
|-1 + 4| -3 = 0 |-7 + 4| - 3 = 0
|-3| - 3 = 0 |-3| - 3 = 0
3-3=0 3-3=0
0=0 0=0

8. EMPTY SETS.
• |3d - 9| + 6 = 0 First isolate the variable by
subtracting 6 from both sides.
|3d - 9| = -6
There is no need to go any further with this
problem!
• Absolute value is never negative.
• Therefore, the solution is the empty set!

9. SOLVE.
• Solve: 3|x - 5| = 12
|x - 5| = 4
x-5=4 and x - 5 = -4
x=9 x=1
• Check: 3|x - 5| = 12
3|9 - 5| = 12 3|1 - 5| = 12
3|4| = 12 3|-4| = 12
3(4) = 12 3(4) = 12
12 = 12 12 = 12

10. SOLVE: 6|5x + 2| = 312
• Isolate the absolute value expression by dividing by 6.
6|5x + 2| = 312
|5x + 2| = 52
• Set up two equations to solve.
5x + 2 = 52 5x + 2 = -52
5x = 50 5x = -54
x = 10 or x = -10.8
•Check: 6|5x + 2| = 312 6|5x + 2| = 312
6|5(10)+2| = 312 6|5(-10.8)+ 2| = 312
6|52| = 312 6|-52| = 312
312 = 312 312 = 312

11. SOLVE: 3|x + 2| -7 = 14
• Isolate the absolute value expression by adding 7 and dividing by 3.
3|x + 2| -7 = 14
3|x + 2| = 21
|x + 2| = 7
• Set up two equations to solve.
x+2=7 x + 2 = -7
x=5 or x = -9
•Check: 3|x + 2| - 7 = 14 3|x + 2| -7 = 14
3|5 + 2| - 7 = 14 3|-9+ 2| -7 = 14
3|7| - 7 = 14 3|-7| -7 = 14
21 - 7 = 14 21 - 7 = 14
14 = 14 14 = 14

12. SOLVE.
Solve: |8 + 5a| = 14 - a
8 + 5a = 14 - a and 8 + 5a = -(14 – a)
Set up your 2 equations, but
make sure to negate the entire
right side of the second
equation.
8 + 5a = 14 - a and 8 + 5a = -14 + a
6a = 6 4a = -22
a=1 a = -5.5
Check: |8 + 5a| = 14 - a
|8 + 5(1)| = 14 - 1 |8 + 5(-5.5) = 14 - (-5.5)
|13| = 13 |-19.5| = 19.5
13 = 13 19.5 = 19.5