STANDARD 3:
ABSOLUTE VALUE EQUATIONS & INEQUALITIES
Absolute Value
Absolute Value is the distance away from zero. The absolute value of a number is always positive because distance is never negative.
Solving Absolute Value Equations
Steps to solving an absolute value equation:
1. Get the absolute value is isolated on one side of the equation
2. Create two equations (positive and negative)
3. Solve each equation
4. Check your answers in the original equation!
Example1: Solve | x + 2 | = 7
- To clear the absolute-value bars, I must split the equation into its two possible two cases, one case for each sign:
- (x + 2) = 7 or –(x + 2) = 7
x + 2 = 7 or –x – 2 = 7
x = 5 or –9 = x
Example2: Solve | 2x – 3 | – 4 = 3
- First, I'll isolate the absolute-value part; that is, I'll get the absolute-value expression by itself on one side of the "equals" sign, with everything else on the other side:
- | 2x – 3 | – 4 = 3
| 2x – 3 | = 7
- (2x – 3) = 7 or –(2x – 3) = 7
2x – 3 = 7 or –2x + 3 = 7
2x = 10 or –2x = 4
x = 5 or x = –2
Solving Absolute Value Inequalities
How to solve an absolute value inequality:
1. Get the absolute value is isolated on one side of the equation
2. Create two inequalities (positive and negative)
3. Solve each inequality *** if you divide or multiply by a negative, flip the inequality sign!!!
4. Put your solution on a number line! * remember that < and > are open circles, while < or > should be graphed with closed circles!
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