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

This page presents a complete guide to simplifying expressions that involve absolute value, with detailed solutions and explanations. You will find examples of numerical and algebraic simplifications using absolute value rules, and square roots. The absolute value properties are listed below to guide your practice.

Properties of Absolute Value

• \( |a| \geq 0 \) for any real number \( a \)
• \( |-a| = |a| \)
• \( |a| = 0 \) if and only if \( a = 0 \)
• \( |ab| = |a||b| \)
• \( \left|\dfrac{a}{b}\right| = \dfrac{|a|}{|b|},\ b \neq 0 \)
• \( |a + b| \leq |a| + |b| \) (Triangle Inequality)
• \( |a - b| \geq ||a| - |b|| \)
• \( \sqrt{a^2} = |a| \)

Problems and Questions

1. Simplify the following numerical expressions with absolute value :

   1. \(|5| =\)
   2. \(|-6| =\)
   3. \(|-7 + 10| =\)
   4. \(|-2 - 9| =\)
   5. \(|10 - 20| - |-7| =\)
   6. \(|\pi - 4| - |2| =\)
   7. \(\left|- \dfrac{5}{3} - \dfrac{1}{2}\right| - \left| \dfrac{7}{9} - \dfrac{3}{2}\right| =\)
   8. \(|-1.5 - 3.9| - |-1.6| =\)

2. Simplify the following algebraic expressions with absolute value:

   1. \(|x^2| =\)
   2. \(|x|^2 =\)
   3. \(||x|| =\)
   4. \(|x^2 + 4| =\)
   5. \(|-x^4 - 9| =\)
   6. \(|-|x| - 2| =\)
   7. \( \left| \dfrac{-x}{3} \right| = \)
   8. \(|-x^2 - y^2 + 2xy| =\)

3. Use absolute value to simplify the expressions:

   1. \(\sqrt{x^2} =\)
   2. \(\sqrt{(x + 3)^2} =\)
   3. \(\sqrt{x^2 + 1 + 2x} =\)
   4. \(\sqrt{ \dfrac{25}{x^2}} =\)

Solutions to the Above Problems and Questions

1. 1. \( |5| = 5 \)
   2. \( |-6| = 6 \)
   3. \( |-7 + 10| = |3| = 3 \)
   4. \( |-2 - 9| = |-11| = 11 \)
   5. \( |10 - 20| - |-7| = |-10| - |-7| = 10 - 7 = 3 \)
   6. \( |\pi - 4| - |2| = | -(\pi - 4)| - 2 = (4 - \pi) - 2 = 2 - \pi \)
   7. \( \left|- \dfrac{5}{3} - \dfrac{1}{2}\right| - \left| \dfrac{7}{9} - \dfrac{3}{2} \right| = \left| - \dfrac{13}{6} \right| - \left| - \dfrac{13}{18} \right| = \dfrac{13}{6} - \dfrac{13}{18} = \dfrac{13}{9} \)
   8. \( |-1.5 - 3.9| - |-1.6| = |-5.4| - 1.6 = 5.4 - 1.6 = 3.8 \)
2. 1. \( |x^2| = x^2 \)   (since \( x^2 \geq 0 \))
   2. \( |x|^2 = (|x|)^2 = x^2 \)
   3. \( ||x|| = |x| \)
   4. \( |x^2 + 4| = x^2 + 4 \)   (since \( x^2 + 4 \geq 0 \))
   5. \( |-x^4 - 9| = -(-x^4 - 9) = x^4 + 9 \)   (since expression inside is always negative)
   6. \( | -|x| - 2| = -( -|x| - 2) = |x| + 2 \)   (since expression inside is always negative)
   7. \( \left| \dfrac{-x}{3} \right| = \dfrac{|-x|}{|3|} = \dfrac{|x|}{3} \)
   8. \( |-x^2 - y^2 + 2xy| = |-(x^2 + y^2 - 2xy)| = |-(x - y)^2| = (x - y)^2 \)
3. 1. \( \sqrt{x^2} = |x| \)
      Explanation: The square root of a square is the absolute value of the base, because \( \sqrt{x^2} \) is always non-negative regardless of the sign of \( x \).
   2. \( \sqrt{(x + 3)^2} = |x + 3| \)
      Explanation: Similar to the previous case, the square root of a square expression like \( (x + 3)^2 \) equals the absolute value of the base \( x + 3 \).
   3. \( \sqrt{x^2 + 1 + 2x} = \sqrt{(x + 1)^2} = |x + 1| \)
      Explanation: The expression under the square root simplifies: \[ x^2 + 1 + 2x = x^2 + 2x + 1 = (x + 1)^2, \] and then \( \sqrt{(x + 1)^2} = |x + 1| \).
   4. \( \sqrt{ \dfrac{25}{x^2} } = \dfrac{5}{|x|} \)
      Explanation: The square root of a fraction is the square root of the numerator over the square root of the denominator: \[ \sqrt{ \dfrac{25}{x^2} } = \dfrac{\sqrt{25}}{\sqrt{x^2}} = \dfrac{5}{|x|}. \] The absolute value in the denominator ensures the result is non-negative.

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