Question

solve the compound inequality and choose the correct answer below.
$-3x - 9 \geq 6$ or $-4x - 3 > 1$

Explanation:

Step1: Solve \(-3x - 9 \geq 6\)

Add 9 to both sides: \(-3x - 9 + 9 \geq 6 + 9\)
Simplify: \(-3x \geq 15\)
Divide both sides by -3 (reverse inequality): \(x \leq -5\)

Step2: Solve \(-4x - 3 > 1\)

Add 3 to both sides: \(-4x - 3 + 3 > 1 + 3\)
Simplify: \(-4x > 4\)
Divide both sides by -4 (reverse inequality): \(x < -1\)

Step3: Analyze "or" compound inequality

The solution to \(x \leq -5\) or \(x < -1\) is all numbers that satisfy either inequality. Since \(x \leq -5\) is a subset of \(x < -1\) (all numbers \(\leq -5\) are also \(< -1\)), the combined solution is \(x \leq -5\) (from the first inequality) and \(x < -1\) (which includes the first solution). Visually, this matches the number line with a closed dot at -5 (for \(x \leq -5\)) and an open dot at -1, with the shaded regions covering left of -5 (including -5) and between -5 and -1 (since \(x < -1\) includes that). The last option (fourth) shows a closed dot at -5 and shading to the left (for \(x \leq -5\)) and shading from -5 to -1 (for \(x < -1\)), combining them.

Answer:

The correct number line is the fourth option (with closed dot at -5, open dot at -1, and shading left of -5 and between -5 and -1).