Question

which number line represents the solution set for the inequality (2x - 6geq6(x - 2)+8)?

Explanation:

Step1: Expand the right - hand side

Expand \(6(x - 2)+8\) using the distributive property \(a(b + c)=ab+ac\).

$$6(x - 2)+8=6x-12 + 8=6x-4$$

The inequality becomes \(2x-6\geq6x - 4\).

Step2: Move the \(x\) terms to one side

Subtract \(2x\) from both sides:

$$2x-6-2x\geq6x - 4-2x$$

$$ - 6\geq4x-4$$

Step3: Move the constant terms to one side

Add \(4\) to both sides:

$$-6 + 4\geq4x-4 + 4$$

$$-2\geq4x$$

Step4: Solve for \(x\)

Divide both sides by \(4\):

$$x\leq-\frac{2}{4}=-\frac{1}{2}=-0.5$$

Answer:

The number line that has a closed circle at \(-0.5\) and the arrow points to the left (the third option in the original problem's number - line choices).