Question

solve the compound inequality and choose the correct answer below.
-6x + 2 ≤ 14 or -2x - 4 > -6

Explanation:

Step1: Solve the first inequality \(-6x + 2 \leq 14\)

Subtract 2 from both sides: \(-6x + 2 - 2 \leq 14 - 2\)
Simplify: \(-6x \leq 12\)
Divide both sides by -6 (and reverse the inequality sign): \(x \geq -2\)

Step2: Solve the second inequality \(-2x - 4 > -6\)

Add 4 to both sides: \(-2x - 4 + 4 > -6 + 4\)
Simplify: \(-2x > -2\)
Divide both sides by -2 (and reverse the inequality sign): \(x < 1\)

Step3: Analyze the compound inequality "or"

The compound inequality is \(x \geq -2\) or \(x < 1\). Since "or" means the union of the two solution sets, and the solution set of \(x \geq -2\) includes all numbers from -2 (inclusive) to infinity, and the solution set of \(x < 1\) includes all numbers less than 1, the union of these two sets is all real numbers. But looking at the number lines, we need to check the endpoints and the direction.

For \(x \geq -2\), the endpoint at -2 is a closed circle (since it's "greater than or equal to"). For \(x < 1\), the endpoint at 1 is an open circle (since it's "less than").

Looking at the options, the first number line has a closed circle at -2 (blue) and an open circle at 1 (red), with the blue line going to the right from -2 and the red line going to the left from 1, covering all real numbers.

Answer:

The first number line (with closed circle at -2, open circle at 1, blue line right from -2, red line left from 1)