Which number line represents the solution set for the inequality -4(x+3) ≤ -2-2x?

Step 1: Define the Problem

Identify the inequality to solve: $-4(x+3) \leq -2 - 2x$.

Step 2: Break Down the Problem

1. Distribute and simplify the left side of the inequality.
2. Move all terms involving $x$ to one side of the inequality.
3. Solve for $x$.

Step 3: Apply Relevant Concepts

1. Distribute $-4$ on the left side: $-4x - 12 \leq -2 - 2x$
2. Add $4x$ to both sides to move terms involving $x$ to one side: $-12 \leq -2 + 2x$
3. Add 2 to both sides to isolate terms involving $x$: $-10 \leq 2x$
4. Divide both sides by 2 to solve for $x$: $-5 \leq x$

Step 4: Analysis, Verify and Summarize

• The inequality simplifies to $x \geq -5$.
• This means the solution set includes all numbers greater than or equal to $-5$.

Final Answer

The number line representing the solution set for the inequality $-4(x+3) \leq -2 - 2x$ is:

A closed circle at $-5$ with a line extending to the right, indicating all numbers greater than or equal to $-5$.