Simplify Absolute Value Expressions

Definition, Rules, and Detailed Solutions

Expressions with absolute value can be simplified only when the sign of the expression inside the absolute value is known. 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. If that quantity is negative, we multiply it by \( -1 \).

NOTE: The absolute value is either positive or equal to zero and is NEVER negative.

Definition & Core Rules

Definition of the Absolute Value Function:

\[ \text{If } x \ge 0 \text{ then } | x | = x \] \[ \text{If } x < 0 \text{ then } | x | = (-1)x = -x \]

For example, \( | 2 | = 2 \) because 2 is positive. However, \( | -5 | = (-1)(-5) = 5 \) because -5 is negative.

Rule Formula Example
Product Rule \( |A \times B| = |A| \times |B| \) \( |(-2)x^2| = |-2| \times |x^2| = 2x^2 \)
Quotient Rule \( \left| \dfrac{A}{B} \right| = \dfrac{|A|}{|B|} \) \( \left| \dfrac{-10}{x^2 + 1} \right| = \dfrac{|-10|}{|x^2 + 1|} = \dfrac{10}{x^2 + 1} \)
Square Root of a Square \( \sqrt{A^2} = |A| \) \( \sqrt{(x^2 + 5)^2} = |x^2 + 5| = x^2 + 5 \)

Example 1: Numerical Expressions

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. \( \left| \dfrac{-2}{5}\right| \)
View Solutions to Example 1
  1. \( -10 \) is negative, and according to the definition: \[ | -10 | = (-1)(-10) = 10 \]
  2. According to the definition: \[ | 0 | = 0 \]
  3. \( -2 + 10 = 8 \), which is positive: \[ | -2 + 10 | = | 8 | = 8 \]
  4. \( 1/2 - 20 = -39/2 \) is negative: \[ | 1/2 - 20 | = | -39/2 | = -(-39/2) = 39/2 \]
  5. \( \sqrt{3} - 5 \approx -3.27 \), which is negative: \[ | \sqrt{3} - 5 | = - (\sqrt{3} - 5) = 5 - \sqrt{3} \]
  6. \( \sqrt{14} - 3 \pi + 10 \approx 4.32 \), which is positive: \[ | \sqrt{14} - 3 \pi + 10 | = \sqrt{14} - 3 \pi + 10 \]
  7. \( \dfrac{-2}{5} = -\dfrac{2}{5} \) is negative: \[ \left| \dfrac{-2}{5} \right| = \left| -\dfrac{2}{5} \right| = (-1) \left(-\dfrac{2}{5}\right) = \dfrac{2}{5} \]

Example 2: Algebraic Expressions

Simplify the algebraic expressions and rewrite them without absolute value:

  1. \( | x^2 + 1 | \)
  2. \( | x + 3 | \), if \( x < -3 \)
  3. \( | -x + 2 | \), if \( x > 2 \)
View Solutions to Example 2
  1. \( x^2 + 1 \) is always positive, and according to the definition: \[ \left| x^2 + 1 \right| = x^2 + 1 \]
  2. If \( x < -3 \), then \( x + 3 < 0 \). According to the definition: \[ \left| x + 3 \right| = - (x + 3) = -x - 3 \]
  3. If \( x > 2 \), then \( x - 2 > 0 \) and \( -x + 2 < 0 \). According to the definition: \[ \left| -x + 2 \right| = -(-x + 2) = x - 2 \]

More Questions on Absolute Value Expressions

Rewrite the following expressions without absolute value or square root symbols:

  1. \( \left| -2 (-19 + 7) \right| = \)
  2. If \( x < 9 \), then \( \left| x - 9 \right| = \)
  3. If \( -3 < x < 3 \), 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. If \( x < 2 \), then \( \sqrt{ x^2 - 4x + 4 } = \)
View Solutions to More Questions
  1. \[ | -2 (-19 + 7) | = | -2 (-12) | = |24| = 24 \]
  2. If \( x < 9 \), then \( x - 9 < 0 \), hence: \[ | x - 9 | = - (x - 9) = -x + 9 \]
  3. If \( -3 < x < 3 \), then \( |x| < 3 \), hence \( x^2 < 9 \). This can be written as \( x^2 - 9 < 0 \), hence: \[ | x^2 - 9 | = - (x^2 - 9) = -x^2 + 9 \]
  4. If \( |x| > 2 \), then \( x^2 > 4 \) and \( x^2 - 4 > 0 \), hence: \[ | x^2 - 4 | = x^2 - 4 \]
  5. Use the quotient rule: \[ \left| \frac{ |-x| }{ x^2 - 1 } \right| = \frac{ ||-x|| }{ |x^2 - 1| } \] 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: \[ |(-x^2 - 4)(-x^4 - 9)| = |-x^2 - 4| \cdot |-x^4 - 9| \] Since both expressions \( (-x^2 - 4) \) and \( (-x^4 - 9) \) are negative: \[ |(-x^2 - 4)(-x^4 - 9)| = (-1)(-x^2 - 4)(-1)(-x^4 - 9) = (x^2 + 4)(x^4 + 9) \]
  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 \( x < 2 \), then \( x - 2 < 0 \), so: \[ |x - 2| = - (x - 2) = -x + 2 \] \[ \sqrt{x^2 - 4x + 4} = |x - 2| = -x + 2 \]

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