Digital SATMath: Advanced MathMedium
A quality control engineer is inspecting a component. The acceptable length L (in millimeters) of the component must satisfy the inequality |L - 15| < 0.5. Which of the following ranges represents the acceptable lengths for the component?
- AL < 14.5 or L > 15.5
- B14.5 < L < 15.5
- CL < 15.5
- DL > 14.5
Show answer & explanationAnswer & explanation
Correct answer: B. 14.5 < L < 15.5
The inequality |L - 15| < 0.5 can be rewritten as a compound inequality: -0.5 < L - 15 < 0.5. To solve for L, add 15 to all parts of the inequality: -0.5 + 15 < L - 15 + 15 < 0.5 + 15. This simplifies to 14.5 < L < 15.5. This range represents all acceptable lengths for the component.
Why the other options are wrong
- A. This represents the solution for |L - 15| > 0.5, not |L - 15| < 0.5.
- C. Only represents the upper bound, neglecting the lower bound.
- D. Only represents the lower bound, neglecting the upper bound.
Solving Absolute Value Inequalities (<)
For an inequality of the form |ax + b| < c (where c > 0), it is equivalent to the compound inequality -c < ax + b < c. This means the expression inside the absolute value is between -c and c.
- Translates to a 'sandwich' inequality.
- Solve by isolating the variable in the middle.
- Represents values 'within' a certain distance from a center point.
Memory trick: Less than means 'between' (like a sandwich)!