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Absolute Value Inequalities quiz

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  • What is the first step when solving an absolute value inequality?
    The first step is to isolate the absolute value expression on one side of the inequality.
  • How do you rewrite the inequality |x| < a, where a > 0, without absolute values?
    You rewrite it as a three-part inequality: -a < x < a.
  • What does the inequality |x| > a mean in terms of two separate inequalities?
    It means x > a or x < -a.
  • How do you solve |x + 1| ≤ 2?
    Rewrite as -2 ≤ x + 1 ≤ 2, then solve for x to get -3 ≤ x ≤ 1.
  • What is the solution to |x + 1| < -1?
    There is no solution because absolute value cannot be less than a negative number.
  • What is the solution to |x + 1| ≤ 0?
    The solution is x + 1 = 0, so x = -1.
  • How do you express the solution -3 ≤ x ≤ 1 in interval notation?
    It is written as [-3, 1].
  • What does a bracket [ ] mean in interval notation for absolute value inequalities?
    A bracket means the endpoint is included in the solution set (less than or equal to, or greater than or equal to).
  • How do you solve |x + 1| ≥ 2?
    Rewrite as x + 1 ≥ 2 or x + 1 ≤ -2, then solve to get x ≥ 1 or x ≤ -3.
  • What is the interval notation for the solution x ≥ 1 or x ≤ -3?
    It is (-∞, -3] ∪ [1, ∞).
  • What happens if you have |x| < 0?
    There is no solution because absolute value cannot be negative.
  • What is the solution to |x| ≥ 0?
    The solution is all real numbers, since absolute value is always at least zero.
  • When solving a three-part inequality, what must you do to all three sides?
    You must perform the same operation to all three sides to maintain the inequality.
  • How do you graph the solution to |x| < a on a number line?
    Shade the region between -a and a, using open circles if the inequality is strict (<) or closed circles if it includes equality (≤).
  • What special case occurs when the number on the other side of the absolute value is negative in an inequality?
    If the inequality is |x| < negative number, there is no solution; if |x| > negative number, the solution is all real numbers.