Question

which number line represents the solution to the inequality 2k - 4 ≥ -22?
(there are four number lines labeled a, b, c, d with different markings and arrows)

Explanation:

Step1: Solve the inequality \(2k - 4\geq - 22\)

Add 4 to both sides of the inequality: \(2k-4 + 4\geq-22 + 4\), which simplifies to \(2k\geq - 18\).

Step2: Divide both sides by 2

Divide each side of the inequality \(2k\geq - 18\) by 2: \(\frac{2k}{2}\geq\frac{-18}{2}\), so \(k\geq - 9\).

Step3: Analyze the number lines

We need a number line that shows all values of \(k\) greater than or equal to - 9. A closed circle at - 9 (since the inequality is "greater than or equal to") and an arrow pointing to the right (for values greater than - 9). Looking at the options, we check the position of the closed circle and the direction. The number line with a closed circle at - 9 and the arrow to the right (we assume from the options, let's check the visual: if we consider the number lines, the one with closed circle at - 9 and shading to the right. Let's re - check the inequality solution: \(k\geq - 9\), so the number line should have a closed dot at - 9 (because the inequality is inclusive) and the line should be shaded to the right (for values greater than - 9). Now, looking at the given number lines (A, B, C, D), we need to find which one has a closed dot at - 9 and the arrow going right. Let's assume the number lines: if we have the number lines with labels A, B, C, D, and we solved \(k\geq - 9\), so the correct number line is the one with a closed circle at - 9 and the arrow pointing to the right (towards positive numbers). Let's check the options again. Let's re - solve the inequality carefully:

1. Start with \(2k-4\geq - 22\)

• Add 4 to both sides: \(2k\geq - 22 + 4=-18\)
• Divide by 2: \(k\geq - 9\)

So the solution is \(k\geq - 9\). So the number line should have a closed circle at \(k = - 9\) (since the inequality is non - strict, \(\geq\)) and the arrow should be to the right (for all \(k\) greater than or equal to - 9). Now, looking at the number lines, we need to identify which one has a closed circle at - 9 and the arrow in the positive direction.

Answer:

The number line with a closed circle at \(-9\) and the arrow pointing to the right (assuming the correct option is the one that represents \(k\geq - 9\), for example, if we consider the labels, let's say if the number line with closed circle at - 9 and arrow right is, for example, if we look at the given image, let's assume the correct one is (we need to match with the visual, but from the inequality solution, the correct number line is the one that represents \(k\geq - 9\), so the answer is the number line with closed circle at - 9 and arrow to the right. If we assume the options, and based on the solution \(k\geq - 9\), the correct number line is the one with closed dot at - 9 and shading to the right. Let's say the correct option is (let's check the original problem's number lines: if we have the number lines, and we solved \(k\geq - 9\), so the correct number line is the one with closed circle at - 9 and the arrow going right. So the answer is the number line (let's assume the label, but from the solution, the correct representation is a closed circle at - 9 and arrow to the right. So the answer is the number line that corresponds to \(k\geq - 9\), which is the one with closed circle at - 9 and arrow right.