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Here are some tips for Absolute Value Equations, which aligns with New York state standards:
Absolute Value Equation
You must understand absolute values and inequalities to master this topic.
To review absolute value, see Absolute Value 1
and Absolute Value 2.
To review the Single Variable Inequalities topic, see here.
| Where A is the expression between the absolute value signs and b is a positive number, | |
| • when solving an equation with the form | A| = b, solve | |
| A = b or A = - b | |
| • when solving an equality with the form | A| < b, solve | |
| - b < A < b | |
| This rule is also true for ≤. | |
| • when solving an equality with the form | A| > b, solve | |
| A < - b or A > b | |
| This rule is also true for ≥. | |
Example 1: Equations
Solve for x.1. |x| = 5
Following the rules above, we get x = 5 or x = -5.2. |x| = -1Answer:
By definition of absolute value, |x| is always positive. So there is no x value that will work in this equation.3. |x - 6| = 4Answer:
x - 6 = 4 or x - 6 = -4 x - 6 + 6 = 4 + 6 x - 6 + 6 = -4 + 6 x = 10 x = 2 Answer:
Example 2: Inequalities
Solve for x.1. |x| ≥ 5
Following the rules above, we get x ≥ 5 or x ≤ -5.2. |7 - x| ≤ 2Answer:
-2 ≤ 7 - x ≤ 2 -2 ≤ 7 - x and 7 - x ≤ 2 -2 - 7 ≤ 7 - x - 7 7 - x - 7 ≤ 2 - 7 -9 ≤ -x -x ≤ -5 9 ≥ x x ≥ 5 Combine the results: 9 ≥ x ≥ 5
This is the same as 5 ≤ x ≤ 9
Answer:3. |8 - x| > 2
8 - x < -2 or 8 - x > 2 8 - x - 8 < -2 - 8 8 - x - 8 > 2 - 8 -x < -10 -x > - 6 x > 10 x < 6 Answer:
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