Simplify Absolute Value Expressions

Definition of the Absolute Value Function

This is a tutorial on how to simplify expressions with absolute value. It is important to understand the definition of the absolute value first.
Expressions with absolute value can be simplified only when the sign of the expression inside the absolute value is known \[ \text{If} \; x \ge 0 \; \text{then} \; | x | = x \] \[ \text{If} \; x \lt 0 \; \text{then} \; | x | = (-1) x = - x \] In order to simplify an expression with absolute value, we examine the sign of the quantity inside the absolute value. If that quantity is positive or equal to zero, its absolute value is the quantity itself.
\( | 2 | = 2 \) because the quantity 2 which inside the abolsute value is positive.
It that quantity is negative, we multiply it by \( -1 \).
\( | - 5 | = (- 1)(- 5) = 5 \) because the quantity \( - 5 \) which is inside the absolute value is negative.
NOTE that the absolute value is either positive or equal to zero and is NEVER negative.

Examples with Detailed Solutions

Example 1

Simplify the expressions and rewrite them without absolute value
  1. \( | -10 | \)
  2. \( | 0 | \)
  3. \( | -2 + 10 | \)
  4. \( | 1/2 -20 | \)
  5. \( | sqrt{3} - 5 | \)
  6. \( | \sqrt{14} - 3 \pi+ 10 | \)
  7. \( \left| \dfrac{-2}{5}\right| \)
Solution to Example1
  1. \( -10 \) is and according to the definition above | - 10 | can be simplified as follows \[ | -10 | = (-1)(-10) = 10 \]
  2. According to the definition above \( | 0 | \) can be simplified as follows: \[ | 0 | = 0 \]
  3. \( - 2 + 10 = 8 \) is positive and according to the definition above | -2 + 10 | can be simplified as follows \[ | -2 + 10 | = | 8 | = 8 \]
  4. \( 1/2 - 20 = -39/2 \) is negative, according to the definition above | 1/2 -20 | can be simplified as follows: \[ | 1/2 - 20 | = | -39/2 | = -(-39/2) = 39/2 \]
  5. \( \sqrt {3} - 5 \approx -3.27 \) which is negative, according to the definition and can be simplified as follows \[ | \sqrt {3} - 5 | = - ( \sqrt {3} - 5) = 5 - \sqrt {3} \]
  6. \( \sqrt{14} - 3 \pi + 10 \approx 4.32 \) which is positive , according to the definition above it can be simplified as follows: \[ | \sqrt{14} - 3 \pi + 10 | = \sqrt{14} - 3 \pi + 10 \]
  7. \( \dfrac{-2}{5} = - \dfrac{2}{5} \) is negative, according to the definition above it can be simplified as follows: \[ | \dfrac{-2}{5} | = | - \dfrac{2}{5} | =(-1) (- \dfrac{2}{5}) = \dfrac{2}{5} \]

Examples with algebraic expressions are now presented.

Example 2

Simplify the algebraic expressions and rewrite tem without absolute value.
  1. \(| x^2 + 1 | \)
  2. \( | x + 3 | \) , if \( x \lt -3 \)
  3. \( | - x + 2 | \) , if \( x \gt 2 \)

Solution to Example 2

  1. \( x^2 + 1 \) is always *positive* and according to the definition above, \( \left| x^2 + 1 \right| \) can be simplified as follows: \[ \left| x^2 + 1 \right| = x^2 + 1 \]
  2. If \( x \lt -3 \), then \( x + 3 \lt 0 \). According to the definition above, \( \left| x + 3 \right| \) can be simplified as follows: \[ \left| x + 3 \right| = - (x + 3) = -x - 3 \]
  3. If \( x > 2 \), then \( x - 2 > 0 \) and \( -x + 2 \lt 0 \). According to the definition above, \( \left| -x + 2 \right| \) can be simplified as follows: \[ \left| -x + 2 \right| = -(-x + 2) = x - 2 \]

Important Rules for the Absolute Value Expressions

  1. Product rule: \( |A \times B| = |A| \times |B| \)

    Example: \[ |(-2)x^2| = |-2| \times |x^2| = 2x^2 \] \( x^2 \) is either positive or zero

  2. Quotient rule: \( \left| \dfrac{A}{B} \right| = \dfrac{|A|}{|B|} \)

    Example: \[ \left| \dfrac{-10}{x^2 + 1} \right| = \dfrac{|-10|}{|x^2 + 1|} = \dfrac{10}{x^2 + 1} \] \( x^2 + 1 \) is positive.

  3. Square root of a square rule: \( \sqrt{A^2} = |A| \)

    Example: \[ \sqrt{(x^2 + 5)^2} = |x^2 + 5| = x^2 + 5 \] \( x^2 + 5 \) is positive.

More Questions on Absolute Value Expressions with Solutions

Rewrite the following expressions without absolute value.
  1. \( \left| -2 (-19 + 7) \right| = \)
  2. If \( x \lt 9 \), then \( \left| x - 9 \right| = \)
  3. If \( -3 \lt x \lt 3 \), then \( \left| x^2 - 9 \right| = \)
  4. If \( \left| x \right| > 2 \), then \( \left| x^2 - 4 \right| = \)
  5. If \( x > 1 \), then \( \left| \dfrac{ |-x| }{ x^2 - 1 } \right| = \)
  6. \( \left| ( - x^2 - 4)( - x^4 - 9) \right| = \)
  7. Rewrite without square root or absolute value. If \( x \lt 2 \), then \[ \sqrt{ x^2 - 4x + 4 } = \]

Solutions to the Above Questions

  1. \[ | -2 (-19 + 7) | = | -2 (-12) | = |24| = 24 \]
  2. If \( x \lt 9 \), then \( x - 9 \lt 0 \), hence \[ | x - 9 | = - (x - 9) = -x + 9 \]
  3. If \( -3 \lt x \lt 3 \), then \( |x| \lt 3 \) , hence \( x^2 \lt 9 \) which can be written: \( x^2 - 9 \lt 0 \), hence \[ | x^2 - 9 | = - (x^2 - 9) = -x^2 + 9 \]
  4. If \( |x| > 2 \), then \( x^2 > 4 \) and \( x^2 - 4 > 0 \), hence \[ | x^2 - 4 | = - (x^2 - 4) = -x^2 + 4 \]
  5. Use the quotient rule to write \[ \left| \frac{ |-x| }{ x^2 - 1 } \right| = \frac{ ||-x|| }{ |x^2 - 1| } \] If \( x > 1 \), then \( x > 0 \), so \( ||-x|| = |x| = x \).
    Also, \( x^2 - 1 > 0 \), so \( |x^2 - 1| = x^2 - 1 \), hence \[ \left| \frac{ |-x| }{ x^2 - 1 } \right| = \frac{x}{x^2 - 1} \]
  6. Use the product rule to write \[ |(-x^2 - 4)(-x^4 - 9)| = |-x^2 - 4| \cdot |-x^4 - 9| \] Since both expressions \( (-x^2 - 4) \) and \( (-x^4 - 9) \) are negative, we have: \[ |(-x^2 - 4)(-x^4 - 9)| = (-1)(-x^2 - 4)(-1)(-x^4 - 9) = (x^2 + 4)(x^4 + 9) \]
  7. Write \( x^2 - 4x + 4 \) as a perfect square: \[ x^2 - 4x + 4 = (x - 2)^2 \] So: \[ \sqrt{x^2 - 4x + 4} = \sqrt{(x - 2)^2} \] Use the square root of a square rule: \[ \sqrt{(x - 2)^2} = |x - 2| \] If \( x \lt 2 \), then \( x - 2 \lt 0 \), so: \[ |x - 2| = - (x - 2) = -x + 2 \] \[ \sqrt{x^2 - 4x + 4} = |x - 2| = -x + 2 \]

More links and references to absolute value functions