Question:
Which of the following inequalities is represented by the number line shown below?
Which of the following inequalities is represented by the number line shown below?
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When an inequality involves a bounded interval like \([-a, b]\), look for an absolute value inequality of the form \(|mx + c| \leq k\).
Updated On: Sep 30, 2025
- \( |4x - 6| \leq 14 \)
- \( |5x - 5| \geq 25 \)
- \( 5x \leq 25 \)
- \( 4x \geq -8 \)
- \( |3x - 12| \leq 6 \)
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The Correct Option is A
Solution and Explanation
Step 1: Recall the form of inequality from number line.
The interval is \(-2 \leq x \leq 5\). This is a bounded inequality that should match an absolute value condition.
Step 2: Check option (A).
\(|4x - 6| \leq 14\) expands to: \[ -14 \leq 4x - 6 \leq 14 \] \[ -8 \leq 4x \leq 20 \quad \Rightarrow \quad -2 \leq x \leq 5 \] This exactly matches the given interval.
Step 3: Check others quickly.
- (B) would give values outside the interval.
- (C) gives \(x \leq 5\), but no lower bound.
- (D) gives \(x \geq -2\), but no upper bound.
- (E) simplifies to \(2 \leq x \leq 6\), which is incorrect.
Hence, only (A) is correct.
Final Answer: \[ \boxed{|4x - 6| \leq 14} \]
The interval is \(-2 \leq x \leq 5\). This is a bounded inequality that should match an absolute value condition.
Step 2: Check option (A).
\(|4x - 6| \leq 14\) expands to: \[ -14 \leq 4x - 6 \leq 14 \] \[ -8 \leq 4x \leq 20 \quad \Rightarrow \quad -2 \leq x \leq 5 \] This exactly matches the given interval.
Step 3: Check others quickly.
- (B) would give values outside the interval.
- (C) gives \(x \leq 5\), but no lower bound.
- (D) gives \(x \geq -2\), but no upper bound.
- (E) simplifies to \(2 \leq x \leq 6\), which is incorrect.
Hence, only (A) is correct.
Final Answer: \[ \boxed{|4x - 6| \leq 14} \]
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