Let's solve the inequalities step-by-step.
Given the compound inequality:
[tex]\[ 2(3x + 5) > 4x - 5 \quad \text{and} \quad 4x - 5 < 3x + 2 \][/tex]
### Step 1: Solve [tex]\( 2(3x + 5) > 4x - 5 \)[/tex]
1. Distribute the [tex]\(2\)[/tex] on the left side:
[tex]\[ 6x + 10 > 4x - 5 \][/tex]
2. Move [tex]\(4x\)[/tex] to the left side and [tex]\(10\)[/tex] to the right side:
[tex]\[ 6x - 4x > -5 - 10 \][/tex]
3. Simplify:
[tex]\[ 2x > -15 \][/tex]
4. Divide by [tex]\(2\)[/tex]:
[tex]\[ x > -\frac{15}{2} \][/tex]
[tex]\[ x > -7.5 \][/tex]
### Step 2: Solve [tex]\( 4x - 5 < 3x + 2 \)[/tex]
1. Move [tex]\(3x\)[/tex] to the left side and [tex]\(-5\)[/tex] to the right side:
[tex]\[ 4x - 3x < 2 + 5 \][/tex]
2. Simplify:
[tex]\[ x < 7 \][/tex]
### Step 3: Combine the solutions
We need [tex]\(x\)[/tex] to satisfy both inequalities [tex]\( x > -7.5 \)[/tex] and [tex]\( x < 7 \)[/tex]. Therefore:
[tex]\[ -7.5 < x < 7 \][/tex]
### Step 4: Check the given values
Now, we need to determine which of the given values [tex]\( 6, 8, -8, -10 \)[/tex] lies within the interval.
- [tex]\(6\)[/tex] lies within the interval [tex]\(-7.5 < x < 7\)[/tex].
- [tex]\(8\)[/tex] does not lie within the interval since [tex]\(8\)[/tex] is greater than [tex]\(7\)[/tex].
- [tex]\(-8\)[/tex] does not lie within the interval since [tex]\(-8\)[/tex] is less than [tex]\(-7.5\)[/tex].
- [tex]\(-10\)[/tex] does not lie within the interval since [tex]\(-10\)[/tex] is less than [tex]\(-7.5\)[/tex].
Thus, the value that [tex]\(x\)[/tex] can take is:
[tex]\[ \boxed{6} \][/tex]
