What are the solutions to |x - 3|

Q: What are the solutions to |x - 3| < 5?

-2 -2 addresses only the lower bound and ignores the upper bound. Both conditions must hold simultaneously, creating an intersection (and), not a union (or). The solution x 8 describes |x - 3| > 5, which is the complement of the correct answer; this represents points more than 5 units from 3. Absolute value inequalities on the CLEP exam come in two forms: |expression| k gives a union of two inequalities (or/union). The geometric interpretation helps: |x - a| k means x is more than k units from a (two rays).

Question

What are the solutions to |x - 3| < 5?

Accepted Answer

-2 < x < 8. The correct answer is -2 < x < 8. The absolute value inequality |x - 3| < 5 means -5 < x - 3 < 5. Adding 3 to all parts: -5 + 3 < x < 5 + 3, giving -2 < x < 8. This represents all numbers within 5 units of 3 on the number line. The solution x < 8 addresses only the upper bound and ignores the lower bound. The solution x > -2 addresses only the lower bound and ignores the upper bound. Both conditions must hold simultaneously, creating an intersection (and), not a union (or). The solution x < -2 or x > 8 describes |x - 3| > 5, which is the complement of the correct answer; this represents points more than 5 units from 3. Absolute value inequalities on the CLEP exam come in two forms: |expression| < k gives a compound inequality (and/intersection), while |expression| > k gives a union of two inequalities (or/union). The geometric interpretation helps: |x - a| < k means x is within k units of a (an interval), while |x - a| > k means x is more than k units from a (two rays).

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