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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.

Answer:

2.  |x| = -1
By definition of absolute value, |x| is always positive. So there is no x value that will work in this equation.

Answer:

3.  |x - 6| = 4
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.

Answer:

2.  |7 - x| ≤ 2
-2 ≤ 7 - x ≤ 2
-27 - x   and   7 - x2
-2 - 77 - x - 7 7 - x - 72 - 7
-9-x -x-5
9x x5
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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