Question

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

Explanation:

Step1: Expand the left side

First, we 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: Move all x terms to one side

Add \(4x\) to both sides of the inequality to get all the \(x\) terms on one side. \(-4x - 12+4x\leq - 2-2x + 4x\), which simplifies to \(-12\leq - 2 + 2x\).

Step3: Move constant terms to the other side

Add \(2\) to both sides of the inequality. \(-12 + 2\leq - 2+2x + 2\), which simplifies to \(-10\leq 2x\).

Step4: Solve for x

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

To represent this on a number line, we have a closed circle at \(-5\) (since the inequality is "greater than or equal to") and the line extends to the right.

Answer:

The number line with a closed circle at \(-5\) and an arrow pointing to the right (towards positive infinity) represents the solution set \(x\geq - 5\).