Question

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

Explanation:

Step1: Expand the left - hand side

Expand $-4(x + 3)$ using the distributive property $a(b + c)=ab+ac$. Here $a=-4$, $b = x$, $c = 3$. So $-4(x + 3)=-4x-12$. The inequality becomes $-4x-12\leq - 2-2x$.

Step2: Add $4x$ to both sides

Adding $4x$ to both sides of the inequality $-4x-12\leq - 2-2x$ gives: $-4x + 4x-12\leq - 2-2x+4x$, which simplifies to $-12\leq - 2 + 2x$.

Step3: Add 2 to both sides

Adding 2 to both sides of $-12\leq - 2 + 2x$ gives: $-12 + 2\leq - 2+2+2x$, which simplifies to $-10\leq 2x$.

Step4: Divide both sides by 2

Dividing both sides of $-10\leq 2x$ by 2 gives: $\frac{-10}{2}\leq\frac{2x}{2}$, so $-5\leq x$ or $x\geq - 5$. This means the solution set includes all real numbers greater than or equal to - 5. On a number - line, this is represented by a closed circle at - 5 and an arrow pointing to the right.

Answer:

The number line with a closed circle at - 5 and an arrow pointing to the right (the second option from the top in the given image).