Solving Absolute Value Equations

What is Absolute Value?

• The absolute value of a number is the number of units it is from zero on the number line.

• 5 and -5 have the same absolute value. • The symbol |x| represents the absolute value

of the number x.

• |-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

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

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

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|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

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

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

• |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.negative.• Therefore, the solution is the Therefore, the solution is the empty setempty set!!

• Solve: 3|x - 5| = 12 |x - 5| = 4x - 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

Solve: |8 + 5a| = 14 - a8 + 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