Question

solve the compound inequality.\\(3x - 6 < -24\\) or \\(4x + 3 \geq 23\\)\
graph the solution on the number line.

Explanation:

Step1: Solve first inequality

Start with $3x - 6 < -24$. Add 6 to both sides:
$3x - 6 + 6 < -24 + 6$
$3x < -18$
Divide by 3:
$\frac{3x}{3} < \frac{-18}{3}$
$x < -6$

Step2: Solve second inequality

Start with $4x + 3 \geq 23$. Subtract 3 from both sides:
$4x + 3 - 3 \geq 23 - 3$
$4x \geq 20$
Divide by 4:
$\frac{4x}{4} \geq \frac{20}{4}$
$x \geq 5$

Step3: Graph instructions

For $x < -6$: Use an open circle at $-6$, draw an arrow pointing left.
For $x \geq 5$: Use a closed circle at $5$, draw an arrow pointing right.

Answer:

The solution to the compound inequality is $x < -6$ or $x \geq 5$.
To graph:

1. Place an open circle at $-6$ on the number line, with an arrow extending left from this point.
2. Place a closed circle at $5$ on the number line, with an arrow extending right from this point.