Question

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

Explanation:

Step1: Solve the first inequality

Solve \(-6x + 2\leq14\).
Subtract \(2\) from both sides: \(-6x\leq14 - 2\), so \(-6x\leq12\).
Divide both sides by \(-6\) (and reverse the inequality sign): \(x\geq - 2\).

Step2: Solve the second inequality

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

Step3: Analyze the compound - inequality

The compound inequality is \(x\geq - 2\) or \(x < 1\). Since \(x\geq - 2\) includes all values from \(-2\) to \(\infty\) and \(x < 1\) includes all values from \(-\infty\) to \(1\), the union of these two sets is all real numbers. But if we consider the number - line representation:
For \(x\geq - 2\), we have a closed circle at \(-2\) (because the inequality is \(\geq\)) and the line goes to the right. For \(x < 1\), we have an open circle at \(1\) (because the inequality is \(<\)) and the line goes to the left. The combined graph has a closed circle at \(-2\), an open circle at \(1\), and the line covers from \(-2\) (including \(-2\)) to \(1\) (not including \(1\)) and beyond in both directions.

Answer:

The fourth option (the last number - line in the given choices, which has a closed circle at \(-2\), an open circle at \(1\), and the line extending from \(-2\) to the right and from \(1\) to the left) is correct.