Question

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

Explanation:

Step1: Expand the left side

Expand \(-4(x + 3)\) using the distributive property \(a(b + c)=ab+ac\). 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 to get the \(x\) - terms on one side. \(-4x + 4x-12\leq-2-2x + 4x\), which simplifies to \(-12\leq - 2 + 2x\).

Step3: Add \(2\) to both sides

Adding \(2\) to both sides: \(-12 + 2\leq-2+2 + 2x\), so \(-10\leq2x\).

Step4: Divide both sides by \(2\)

Divide both sides by \(2\): \(\frac{-10}{2}\leq\frac{2x}{2}\), which gives \(x\geq - 5\).

Answer:

The number line with a closed - circle at \(-5\) and a blue line extending to the right (the first option among the given number - line choices) represents the solution set \(x\geq - 5\).