Questions: Hana wrote the following inequality: 3x + 2 ≤ 5(x - 4) What is the solution set to the inequality she wrote? A. x ≤ 3 B. x ≥ 3 C. x ≤ -11 D. x ≥ 11

Hana wrote the following inequality: 3x + 2 ≤ 5(x - 4) What is the solution set to the inequality she wrote? A. x ≤ 3 B. x ≥ 3 C. x ≤ -11 D. x ≥ 11
Transcript text: 1. Hana wrote the following inequality: \[ 3 x+2 \leq 5(x-4) \] What is the solution set to the inequality she wrote? A. $x \leq 3$ B. $x \geq 3$ C. $x \leq-11$ D. $x \geq 11$
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Solution

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Solution Steps

Step 1: Distribute and Simplify the Right Side

Start by distributing the \(5\) on the right side of the inequality:

\[ 3x + 2 \leq 5(x - 4) \implies 3x + 2 \leq 5x - 20 \]

Step 2: Move Terms Involving \(x\) to One Side

Subtract \(3x\) from both sides to get all terms involving \(x\) on one side:

\[ 2 \leq 5x - 3x - 20 \implies 2 \leq 2x - 20 \]

Step 3: Isolate the Variable

Add \(20\) to both sides to isolate the term with \(x\):

\[ 2 + 20 \leq 2x \implies 22 \leq 2x \]

Divide both sides by \(2\) to solve for \(x\):

\[ \frac{22}{2} \leq x \implies 11 \leq x \]

This can be rewritten as:

\[ x \geq 11 \]

Final Answer

The solution set to the inequality is:

\[ \boxed{x \geq 11} \]

Thus, the correct answer is D.

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