To solve the inequality [tex]\(-13 < 3v - 10 \leq 8\)[/tex], we can break it down into two separate inequalities and solve each part for [tex]\(v\)[/tex].
### Step 1: Solve the first part of the inequality
[tex]\[
-13 < 3v - 10
\][/tex]
1. Add 10 to both sides:
[tex]\[
-13 + 10 < 3v - 10 + 10
\][/tex]
[tex]\[
-3 < 3v
\][/tex]
2. Divide both sides by 3:
[tex]\[
\frac{-3}{3} < \frac{3v}{3}
\][/tex]
[tex]\[
-1 < v
\][/tex]
So the solution to the first part is:
[tex]\[
v > -1
\][/tex]
### Step 2: Solve the second part of the inequality
[tex]\[
3v - 10 \leq 8
\][/tex]
1. Add 10 to both sides:
[tex]\[
3v - 10 + 10 \leq 8 + 10
\][/tex]
[tex]\[
3v \leq 18
\][/tex]
2. Divide both sides by 3:
[tex]\[
\frac{3v}{3} \leq \frac{18}{3}
\][/tex]
[tex]\[
v \leq 6
\][/tex]
### Step 3: Combine the two parts into a compound inequality
The solutions from both parts can now be combined:
[tex]\[
-1 < v \leq 6
\][/tex]
So the compound inequality describing all possible values of [tex]\(v\)[/tex] is:
[tex]\[
-1 < v \leq 6
\][/tex]
This means that [tex]\(v\)[/tex] can take any value greater than [tex]\(-1\)[/tex] but less than or equal to [tex]\(6\)[/tex].
