Question

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

Explanation:

Step1: Expand the left - hand side

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

Step2: Add \(4x\) to both sides

Add \(4x\) to both sides of the inequality \(-4x-12\leqslant-2 - 2x\) to get \(-4x + 4x-12\leqslant-2-2x + 4x\).
Simplify to \(-12\leqslant-2 + 2x\).

Step3: Add \(2\) to both sides

Add \(2\) to both sides: \(-12 + 2\leqslant-2+2 + 2x\).
Simplify to \(-10\leqslant2x\).

Step4: Divide both sides by \(2\)

Divide both sides of \(-10\leqslant2x\) by \(2\). Using the rule \(\frac{a}{c}\leqslant\frac{b}{c}\) when \(c>0\), we have \(\frac{-10}{2}\leqslant\frac{2x}{2}\).
Simplify to \(-5\leqslant x\) or \(x\geqslant - 5\).

Answer:

The number line with a closed circle at \(-5\) and a blue arrow pointing to the right (the first option in the given number - line choices) represents the solution set \(x\geqslant - 5\).