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

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. 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