Question

Find the range of \(x\) satisfying both: \(|2x - 7|<5\) and \(x + 3>0\)

• A. \(1<x<6\)
• B. \(x>1\)
• C. \(1<x<7\)
• D. \(x>-3\)

Hint:

When finding the intersection of two or more inequalities, it's often helpful to visualize the solution sets on a number line. This makes it easy to see the overlapping region that represents the final answer.

Answer 1

The Correct Option is A

Step 1: Split on the sign of \( 2x-7 \). Case A: \( x\ge3.5 \), so \( |2x-7|=2x-7 \), and the inequality becomes \( 2x-7<5 \Rightarrow x<6 \). Combined with \( x\ge3.5 \): \( 3.5\le x<6 \).
Step 2: Case B: \( x<3.5 \), so \( |2x-7|=7-2x \), and the inequality becomes \( 7-2x<5 \Rightarrow x>1 \). Combined with \( x<3.5 \): \( 1<x<3.5 \).
Step 3: Union of both cases:
\[ 1<x<6. \]
Step 4: This already satisfies \( x>-3 \), so the final answer is unchanged:
\[ \boxed{1<x<6} \]