Digital SATMath: Advanced MathHard
A quality control engineer is inspecting a component. The acceptable length L (in millimeters) of the component must satisfy the inequality |2L - 10| ≤ 4. What is the range of acceptable lengths for the component?
- AL ≤ 7
- BL ≥ 3
- C3 ≤ L ≤ 7
- DL ≤ 3 or L ≥ 7
Show answer & explanationAnswer & explanation
Correct answer: C. 3 ≤ L ≤ 7
The inequality |2L - 10| ≤ 4 can be rewritten as a compound inequality: -4 ≤ 2L - 10 ≤ 4. Add 10 to all parts: -4 + 10 ≤ 2L ≤ 4 + 10, which simplifies to 6 ≤ 2L ≤ 14. Divide all parts by 2: 6/2 ≤ L ≤ 14/2, resulting in 3 ≤ L ≤ 7.
Why the other options are wrong
- A. This is an incomplete solution, only showing the upper bound.
- B. This is an incomplete solution, only showing the lower bound.
- D. This represents an 'OR' case (greater than), not an 'AND' case (less than or equal to).
Solving Absolute Value Inequalities (Less Than)
An absolute value inequality of the form |ax + b| ≤ c (where c > 0) can be rewritten as a compound inequality: -c ≤ ax + b ≤ c. This represents an 'AND' condition.
- For |X| ≤ c, it means -c ≤ X ≤ c.
- For |X| < c, it means -c < X < c.
- The solution is a single interval on the number line.
Memory trick: Sandwich the Inside, then Solve!