Question

solve the compound inequality.
3x + 6 ≥ 0 and 4x + 5 < 9
graph the solution on the number line.

Explanation:

Step1: Solve \( 3x + 6 \geq 0 \)

Subtract 6 from both sides: \( 3x + 6 - 6 \geq 0 - 6 \)
Simplify: \( 3x \geq -6 \)
Divide both sides by 3: \( \frac{3x}{3} \geq \frac{-6}{3} \)
Result: \( x \geq -2 \)

Step2: Solve \( 4x + 5 < 9 \)

Subtract 5 from both sides: \( 4x + 5 - 5 < 9 - 5 \)
Simplify: \( 4x < 4 \)
Divide both sides by 4: \( \frac{4x}{4} < \frac{4}{4} \)
Result: \( x < 1 \)

Step3: Find the intersection

The solution to the compound inequality "and" is the overlap of \( x \geq -2 \) and \( x < 1 \), so \( -2 \leq x < 1 \)

Answer:

The solution to the compound inequality is \( -2 \leq x < 1 \). For the number line graph, place a closed circle at -2 (since \( x \) can equal -2) and an open circle at 1 (since \( x \) is less than 1 but not equal to 1), then shade the region between -2 and 1.