1. Absolute value equations can be solved by isolating the absolute value expression and setting up two separate equations, one with the expression unchanged and one with the expression negated. 2. The document provides examples of solving various absolute value equations including |x + 8| = 3, |y + 4| - 3 = 0, and 6|5x + 2| = 312. 3. Key steps are to isolate the absolute value expression, set up the two equations, solve each equation separately, and check the solutions.

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