Questions: Solve the inequality. 3x + 2 < 3(x - 6) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is . (Type your answer in interval notation.) B. The solution set is ∅.

Solve the inequality. 3x + 2 < 3(x - 6) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is . (Type your answer in interval notation.) B. The solution set is ∅.
Transcript text: Solve the inequality. \[ 3 x+2<3(x-6) \] Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is $\square$ . (Type your answer in interval notation.) B. The solution set is $\varnothing$.
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Solution

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Solution Steps

Step 1: Expand the right-hand side of the inequality

The given inequality is: \[ 3x + 2 < 3(x - 6) \] First, expand the right-hand side: \[ 3(x - 6) = 3x - 18 \] So the inequality becomes: \[ 3x + 2 < 3x - 18 \]

Step 2: Simplify the inequality

Subtract \(3x\) from both sides to eliminate the \(3x\) terms: \[ 3x + 2 - 3x < 3x - 18 - 3x \] This simplifies to: \[ 2 < -18 \]

Step 3: Analyze the result

The inequality \(2 < -18\) is false. This means there is no value of \(x\) that satisfies the original inequality.

Final Answer

The solution set is \(\varnothing\). Therefore, the correct choice is: \[ \boxed{\text{B. The solution set is } \varnothing} \]

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