Absolute Value Inequalities Worksheets with Answer Key - Free Printable

To solve these absolute value inequalities, we need to isolate the absolute value expression and then set up two separate inequalities based on the definition of absolute value.

General Rules:
1. Isolate the Absolute Value: Move all other numbers to the other side of the inequality.
2. Split the Inequality:
* If the inequality is "greater than" ($>$ or $\ge$), it becomes an "OR" statement. The expression inside is either greater than the positive number OR less than the negative number.
* If the inequality is "less than" ($<$ or $\le$), it becomes an "AND" statement. The expression inside is between the negative and positive numbers.

Let's go through each problem step-by-step.

Problem 1: $|n + 10| > 19$
* The absolute value is already isolated.
* Since it is "greater than", we split it into two parts:
* $n + 10 > 19$ OR $n + 10 < -19$
* Solve the first part: $n > 19 - 10 \rightarrow n > 9$
* Solve the second part: $n < -19 - 10 \rightarrow n < -29$
* Solution: $n > 9$ or $n < -29$

Problem 2: $3|10m + 4| - 1 \ge 41$
* Add 1 to both sides: $3|10m + 4| \ge 42$
* Divide by 3: $|10m + 4| \ge 14$
* Since it is "greater than or equal to", split it:
* $10m + 4 \ge 14$ OR $10m + 4 \le -14$
* Solve first part: $10m \ge 10 \rightarrow m \ge 1$
* Solve second part: $10m \le -18 \rightarrow m \le -1.8$
* Solution: $m \ge 1$ or $m \le -1.8$

Problem 3: $|x - 4| > |4 - x|$
* Notice that $|x - 4|$ is the same as $|4 - x|$. The distance from $x$ to 4 is the same as the distance from 4 to $x$.
* So the inequality becomes $A > A$.
* A number cannot be strictly greater than itself.
* Solution: No Solution

Problem 4: $3|\frac{1}{2}x + 2| + 6 < 15$
* Subtract 6 from both sides: $3|\frac{1}{2}x + 2| < 9$
* Divide by 3: $|\frac{1}{2}x + 2| < 3$
* Since it is "less than", we set up a compound inequality (between):
* $-3 < \frac{1}{2}x + 2 < 3$
* Subtract 2 from all parts: $-5 < \frac{1}{2}x < 1$
* Multiply all parts by 2: $-10 < x < 2$
* Solution: $-10 < x < 2$

Problem 5: $-5|2x + 2| - 3 \ge -3$
* Add 3 to both sides: $-5|2x + 2| \ge 0$
* Divide by -5. Remember: When dividing by a negative number, flip the inequality sign.
* $|2x + 2| \le 0$
* The absolute value of a number represents distance, which is always non-negative (0 or positive).
* The only time an absolute value is less than or equal to 0 is if it is exactly equal to 0.
* So, $2x + 2 = 0$
* $2x = -2 \rightarrow x = -1$
* Solution: $x = -1$

Problem 6: $|k| - 6 \le -1$
* Add 6 to both sides: $|k| \le 5$
* Since it is "less than or equal to", set up a compound inequality:
* $-5 \le k \le 5$
* Solution: $-5 \le k \le 5$

Problem 7: $-3 - 4|-2x - 5| \ge -7$
* Add 3 to both sides: $-4|-2x - 5| \ge -4$
* Divide by -4. Flip the sign:
* $|-2x - 5| \le 1$
* Set up compound inequality:
* $-1 \le -2x - 5 \le 1$
* Add 5 to all parts: $4 \le -2x \le 6$
* Divide by -2. Flip the signs again:
* $-2 \ge x \ge -3$
* Rewrite in standard order: $-3 \le x \le -2$
* Solution: $-3 \le x \le -2$

Problem 8: $-10 + \frac{1}{2}|x - 4| \ge -10$
* Add 10 to both sides: $\frac{1}{2}|x - 4| \ge 0$
* Multiply by 2: $|x - 4| \ge 0$
* Absolute value is always greater than or equal to 0 for any real number.
* Solution: All Real Numbers

Problem 9: $2|k + 9| \ge 28$
* Divide by 2: $|k + 9| \ge 14$
* Split into "OR" statement:
* $k + 9 \ge 14$ OR $k + 9 \le -14$
* Solve first part: $k \ge 5$
* Solve second part: $k \le -23$
* Solution: $k \ge 5$ or $k \le -23$

Problem 10: $|4k - 10| - 8 > 18$
* Add 8 to both sides: $|4k - 10| > 26$
* Split into "OR" statement:
* $4k - 10 > 26$ OR $4k - 10 < -26$
* Solve first part: $4k > 36 \rightarrow k > 9$
* Solve second part: $4k < -16 \rightarrow k < -4$
* Solution: $k > 9$ or $k < -4$

Final Answer:
1. $n > 9$ or $n < -29$
2. $m \ge 1$ or $m \le -1.8$
3. No Solution
4. $-10 < x < 2$
5. $x = -1$
6. $-5 \le k \le 5$
7. $-3 \le x \le -2$
8. All Real Numbers
9. $k \ge 5$ or $k \le -23$
10. $k > 9$ or $k < -4$