Intermediate Algebra Practice: Sample Set 10
Absolute Value Inequalities with Solutions

This set of intermediate algebra problems focuses on solving inequalities involving absolute value. Each question is followed by a detailed solution provided at the bottom of the page. These exercises can be used alongside the tutorial on How to Solve Inequalities with Absolute Value to reinforce understanding and practice key skills.

1. Solve the following absolute value inequalities.
   1. \( |x| \leq 3 \)
   2. \( |x| \geq 5 \)
   3. \( |x| \leq 0 \)
   4. \( |x| \lt 0 \)
   5. \( |x| \geq 0 \)
   6. \( |x| > 0 \)
   7. \( |2x + 2| \leq 10 \)
   8. \( |3x - 7| \leq 0 \)
   9. \( |-x + 3| \geq 7 \)
   10. \( |3x + 9| > 0 \)
   11. \( 2 |4x - 10| + 3 \geq 17 \)
   12. \( -3 |-5x + 2| + 5 < -10 \)

1. Solutions with detailed explanations:
   1. \( |x| \leq 3 \)
      Solution: \( x \in [-3, 3] \)
      Because the absolute value of \(x\) is at most 3, \(x\) lies between \(-3\) and \(3\).
   2. \( |x| \geq 5 \)
      Solution: \( x \in (-\infty, -5] \cup [5, +\infty) \)
      Values of \(x\) are those whose distance from zero is at least 5, so either less than or equal to \(-5\) or greater than or equal to \(5\).
   3. \( |x| \leq 0 \)
      Solution: \( x = \{0\} \)
      The absolute value is zero only at \(x=0\).
   4. \( |x| \lt 0 \)
      Solution: No solutions
      Absolute value is never negative, so no \(x\) satisfies this.
   5. \( |x| \geq 0 \)
      Solution: \( x \in (-\infty, +\infty) \)
      Absolute value is always non-negative, so all real numbers satisfy this.
   6. \( |x| > 0 \)
      Solution: \( x \in (-\infty, 0) \cup (0, +\infty) \)
      All real numbers except zero have positive absolute value.
   7. \( |2x + 2| \leq 10 \)
      Solution: \( x \in [-6, 4] \)
      The expression inside absolute value lies between \(-10\) and \(10\), so \( -10 \leq 2x + 2 \leq 10 \).
   8. \( |3x - 7| \leq 0 \)
      Solution: \( x = \left\{\dfrac{7}{3}\right\} \)
      Absolute value equals zero only if the inside is zero.
   9. \( |-x + 3| \geq 7 \)
      Solution: \( x \in (-\infty, -4] \cup [10, +\infty) \)
      The inside expression satisfies either \( -x + 3 \leq -7 \) or \( -x + 3 \geq 7 \).
   10. \( |3x + 9| > 0 \)
       Solution: \( x \in (-\infty, -3) \cup (-3, +\infty) \)
       All real \(x\) except \(x = -3\), where the inside equals zero.
   11. \( 2|4x - 10| + 3 \geq 17 \)
       Solution: \( x \in (-\infty, \dfrac{3}{4}] \cup \left[\dfrac{17}{4}, +\infty\right) \)
       First isolate the absolute value, then solve the inequalities.
   12. \( -3|-5x + 2| + 5 \lt -10 \)
       Solution: \( x \in (-\infty, -\dfrac{3}{5}) \cup \left(\dfrac{7}{5}, +\infty\right) \)
       Rewrite the inequality to isolate the absolute value and solve accordingly.