Printable math worksheet for identifying solutions to absolute value inequalities.

Here are the step-by-step solutions for each inequality problem on the worksheet.

1) $|X| \ge 12$
* Logic: The absolute value represents distance from zero. We want numbers whose distance is greater than or equal to 12. This means $X$ can be 12 or larger (positive side), or -12 or smaller (negative side).
* Inequality: $X \le -12$ or $X \ge 12$.
* Interval Notation: $(-\infty, -12] \cup [12, \infty)$.
* Match: Option a.

2) $-|X| \le 23$
* Logic: First, isolate the absolute value by dividing both sides by -1. Remember to flip the inequality sign when dividing by a negative number.
* $|X| \ge -23$.
* Analysis: An absolute value is always positive or zero. Since any positive number is automatically greater than -23, this statement is true for every real number.
* Interval Notation: $(-\infty, \infty)$.
* Match: Option c $(-\infty, 23] \cap [-23, \infty)$ represents the overlap of two sets that covers the entire number line.

3) $|X| > 5$
* Logic: We want numbers strictly further away from zero than 5.
* Inequality: $X < -5$ or $X > 5$.
* Interval Notation: $(-\infty, -5) \cup (5, \infty)$.
* Match: Option d.

4) $|X| < 30$
* Logic: We want numbers strictly closer to zero than 30. This creates a range between -30 and 30.
* Inequality: $-30 < X < 30$.
* Interval Notation: $(-30, 30)$.
* Match: Option c.

5) $|X| < -26$
* Logic: Absolute value measures distance, so it can never be negative. It is impossible for a distance to be less than a negative number.
* Result: There is no solution.
* Match: Option d.

6) $|X| > 9$
* Logic: We want numbers strictly further away from zero than 9.
* Inequality: $X < -9$ or $X > 9$.
* Interval Notation: $(-\infty, -9) \cup (9, \infty)$.
* Match: Option b.

7) $-|X| \le 35$
* Logic: Divide by -1 and flip the sign: $|X| \ge -35$.
* Analysis: Since absolute value is always non-negative, it is always greater than or equal to -35. This is true for all real numbers.
* Interval Notation: $(-\infty, \infty)$.
* Match: Option a $(-\infty, 35] \cap [-35, \infty)$ represents the intersection covering the whole number line.

8) $|X| \ge 4$
* Logic: We want numbers whose distance is greater than or equal to 4.
* Inequality: $X \le -4$ or $X \ge 4$.
* Interval Notation: $(-\infty, -4] \cup [4, \infty)$.
* Match: Option a.

9) $|X| > 27$
* Logic: We want numbers strictly further away from zero than 27.
* Inequality: $X < -27$ or $X > 27$.
* Interval Notation: $(-\infty, -27) \cup (27, \infty)$.
* Match: Option c.

10) $-|X| < 16$
* Logic: Divide by -1 and flip the sign: $|X| > -16$.
* Analysis: Absolute value is always positive (or zero). Any positive number is greater than -16. Therefore, this is true for all real numbers.
* Interval Notation: $(-\infty, \infty)$.
* Match: Option c $(-\infty, 16) \cap (-16, \infty)$ represents the overlap covering the entire number line.

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Final Answer:
1) a
2) c
3) d
4) c
5) d
6) b
7) a
8) a
9) c
10) c