To solve the equation [tex]\( \frac{5v + 4}{3} = \frac{3n}{2} \)[/tex] for a specific value of [tex]\( w \)[/tex], we need to determine the value of [tex]\( w \)[/tex] that can replace both [tex]\( v \)[/tex] and [tex]\( n \)[/tex] and make the equation true.
Let's start by running through the options provided: -8, -2, -1, and 8. We will substitute each value into the equation for both [tex]\( v \)[/tex] and [tex]\( n \)[/tex] and check if the equation holds.
### Step-by-Step Evaluation:
1. Option A: [tex]\( w = -8 \)[/tex]
- Substitute [tex]\( v = -8 \)[/tex] and [tex]\( n = -8 \)[/tex] into the equation:
[tex]\[
\frac{5(-8) + 4}{3} = \frac{3(-8)}{2}
\][/tex]
- Simplify both sides:
[tex]\[
\frac{-40 + 4}{3} = \frac{-24}{2}
\][/tex]
[tex]\[
\frac{-36}{3} = -12
\][/tex]
[tex]\[
-12 = -12 \quad \text{(True)}
\][/tex]
- Therefore, when [tex]\( w = -8 \)[/tex], the equation holds true.
2. Option B: [tex]\( w = -2 \)[/tex]
- Substitute [tex]\( v = -2 \)[/tex] and [tex]\( n = -2 \)[/tex]:
[tex]\[
\frac{5(-2) + 4}{3} = \frac{3(-2)}{2}
\][/tex]
- Simplify both sides:
[tex]\[
\frac{-10 + 4}{3} = \frac{-6}{2}
\][/tex]
[tex]\[
\frac{-6}{3} = -3
\][/tex]
[tex]\[
-2 \neq -3 \quad \text{(False)}
\][/tex]
- Hence, [tex]\( w = -2 \)[/tex] does not satisfy the equation.
3. Option C: [tex]\( w = -1 \)[/tex]
- Substitute [tex]\( v = -1 \)[/tex] and [tex]\( n = -1 \)[/tex]:
[tex]\[
\frac{5(-1) + 4}{3} = \frac{3(-1)}{2}
\][/tex]
- Simplify both sides:
[tex]\[
\frac{-5 + 4}{3} = \frac{-3}{2}
\][/tex]
[tex]\[
\frac{-1}{3} = -1.5 \quad \text{(False)}
\][/tex]
- Hence, [tex]\( w = -1 \)[/tex] does not satisfy the equation.
4. Option D: [tex]\( w = 8 \)[/tex]
- Substitute [tex]\( v = 8 \)[/tex] and [tex]\( n = 8 \)[/tex]:
[tex]\[
\frac{5(8) + 4}{3} = \frac{3(8)}{2}
\][/tex]
- Simplify both sides:
[tex]\[
\frac{40 + 4}{3} = \frac{24}{2}
\][/tex]
[tex]\[
\frac{44}{3} = 12 \quad \text{(False)}
\][/tex]
- Hence, [tex]\( w = 8 \)[/tex] does not satisfy the equation.
### Conclusion:
After evaluating all the options, only [tex]\( w = -8 \)[/tex] makes the equation [tex]\( \frac{5v + 4}{3} = \frac{3n}{2} \)[/tex] true.
Therefore, the correct answer is:
[tex]\[
\boxed{-8}
\][/tex]
