To solve the compound inequality, we'll address each inequality separately, then combine the results.
1. Let's solve the first inequality:
[tex]\[
4x > -16
\][/tex]
We need to isolate [tex]\( x \)[/tex]. To do this, we divide both sides of the inequality by 4:
[tex]\[
\frac{4x}{4} > \frac{-16}{4} \implies x > -4
\][/tex]
2. Next, let's solve the second inequality:
[tex]\[
6x \leq -48
\][/tex]
Again, we isolate [tex]\( x \)[/tex] by dividing both sides by 6:
[tex]\[
\frac{6x}{6} \leq \frac{-48}{6} \implies x \leq -8
\][/tex]
Now, we need to combine the results of the two inequalities. The compound inequality is:
[tex]\[
4x > -16 \quad \text{or} \quad 6x \leq -48
\][/tex]
We have determined that:
[tex]\[
x > -4 \quad \text{or} \quad x \leq -8
\][/tex]
Combining these, we see the solution is in two parts:
- [tex]\( x > -4 \)[/tex]
- [tex]\( x \leq -8 \)[/tex]
So, the solution to the compound inequality is:
[tex]\[
x > -4 \quad \text{or} \quad x \leq -8
\][/tex]
In interval notation, the solution is:
[tex]\[
(-\infty, -8] \cup (-4, \infty)
\][/tex]
This is because [tex]\( x \)[/tex] can be any value greater than [tex]\(-4\)[/tex] or any value less than or equal to [tex]\(-8\)[/tex].
