Section 7.2 Solving Absolute Value Equations

Def. Absolute value represents the distance a number is from 0. Thus, it is always positive. Absolute value is denoted by the bars |x|. 5 and -5 have the same absolute value. The symbol |x| represents the absolute value of the number x. |8| = 8 and |-8| = 8

Examples: Ex1:Ex2:Ex3:

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| =26

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.

Example: When solving an equation, isolate the absolute value expression first. Rewrite the equation as two separate equations. 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| | = 3 |3| = 3 |-3| = 3 3 = 3 3 = 3

Examples: Ex4:Ex5:

Special Notes: | 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! negative. Absolute value is never negative. Therefore, the solution is the empty set! Therefore, the solution is the empty set!

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 = 525x + 2 = -52 5x = 50 5x = -54 x = 10orx = 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

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