Question

which number line represents the solution set for the inequality $2x - 6 \geq 6(x - 2) + 8$?
options with number lines (omitted image details for ocr text, retaining the question text and the fact that there are multiple choice options with number lines).

Explanation:

Step1: Simplify the inequality

First, expand the right - hand side of the inequality \(2x - 6\geq6(x - 2)+8\).
Using the distributive property \(a(b + c)=ab+ac\), we have \(6(x - 2)=6x-12\). So the inequality becomes \(2x-6\geq6x - 12 + 8\).
Simplify the right - hand side: \(6x-12 + 8=6x-4\). So the inequality is \(2x-6\geq6x - 4\).

Step2: Move the terms with x to one side

Subtract \(2x\) from both sides of the inequality: \(2x-6-2x\geq6x - 4-2x\).
This simplifies to \(-6\geq4x - 4\).

Step3: Isolate the term with x

Add 4 to both sides of the inequality: \(-6 + 4\geq4x-4 + 4\).
Simplify: \(-2\geq4x\).

Step4: Solve for x

Divide both sides of the inequality by 4: \(\frac{-2}{4}\geq\frac{4x}{4}\).
Simplify: \(-\frac{1}{2}\geq x\) or \(x\leq-\frac{1}{2}\) (which is \(x\leq - 0.5\)).

On a number line, the solution \(x\leq - 0.5\) is represented by a closed dot at \(x=-0.5\) (since the inequality is "less than or equal to") and an arrow pointing to the left (towards negative infinity). Looking at the options, the third option (the one with the closed dot at - 0.5 and the arrow to the left) represents the solution set.

Answer:

The third option (the number line with a closed dot at \(-0.5\) and the arrow pointing to the left, passing through \(-1.5\), \(-1\), \(-0.5\) etc.)