Intermediate Algebra Problems With Answers -
Sample 8 - Absolute Value Expressions

Algebra problems, related to simplifying expressions with absolute value, are presented along with their answers. The solutions are at the bottom of the page. These questions may be used with the tutorials on Simplifying Absolute Value Expressions and exploring Absolute Value Functions.

1. Simplify the following absolute value expressions
   a) | 5 | =
   b) | -6 | =
   c) | -7 + 10 | =
   d) | -2 - 9 | =
   e) | 10 - 20 | - | -7 | =
   f) | π - 4 | - | 2 | =
   g) | -5/3 - 1/2 | - | 7/9 - 1 1/2 | =
   h) | -1.5 - 3.9 | - | -1.6 | =

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2. Simplify the following absolute value expressions
   a) | x ² | =
   b) | x | ² =
   c) | | x | | =
   d) | x ² + 4 | =
   e) | -x ⁴ - 9 | =
   f) | -| x | - 2 | =
   g) | -x / 3 | =
   h) | -x ² - y ² + 2xy| =

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3. Use absolute value to simplify the expressions
   a) √ (x ²) =
   b) √ (x + 3) ²) =
   c) √ (x ² + 1 + 2x) =
   d) √ ( 25 / x ²) =

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1. a) | 5 | = 5
   b) | -6 | = 6
   c) | -7 + 10 | = | 3 | = 3
   d) | -2 - 9 | = | -11 | = 11
   e) | 10 - 20 | - | -7 | = | -10 | - | -7 | = 10 - 7 = 3
   f) | π - 4 | - | 2 | = - (π - 4) - 2 = 2 - π
   g) | -5/3 - 1/2 | - | 7/9 - 1 1/2 | = | -13/6 | - | -13/8 | = 13/6 + 13/8 = 3 19/24
   h) | -1.5 - 3.9 | - | -1.6 | = | -5.4 | - | -1.6 | = 5.4 - 1.6 = 3.8

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2. a) | x ² | = x ²
   b) | x | ² = x ²
   c) | | x | | = | x |
   d) | x ² + 4 | = x ² + 4
   e) | -x ⁴ - 9 | = x ⁴ + 9
   f) | -| x | - 2 | = | x | + 2
   g) | -x / 3 | = | x | / 3
   h) | -x ² - y ² + 2xy| = | -(x ² + y ² - 2xy) | = | -(x - y) ² | = (x - y) ²

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3. a) √ (x ²) = | x |
   b) √ (x + 3) ²) = | x + 3 |
   c) √ (x ² + 1 + 2x) = √ (x + 1) ² = |x + 1|
   d) √ ( 25 / x ²) = 5 / | x |

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