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.

Absolute Value can also be defined as : |-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 6|5x + 2| = 312 |5x + 2| = 52 5x + 2 = 52 5x + 2 = -52 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 3|x + 2| -7 = 14 3|x + 2| = 21 |x + 2| = 7 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

Absolute value is never negative. |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!

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

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