To find the solution to the given inequalities, we need to solve each inequality step-by-step and then identify which of the given options satisfies both inequalities.
Here's the first inequality:
[tex]\[ 5w + 7 > 2 \][/tex]
1. Subtract 7 from both sides to isolate the variable term:
[tex]\[ 5w + 7 - 7 > 2 - 7 \][/tex]
[tex]\[ 5w > -5 \][/tex]
2. Divide both sides by 5:
[tex]\[ w > -1 \][/tex]
Now, let's solve the second inequality:
[tex]\[ 6w - 15 \leq 3(-1 + w) \][/tex]
1. Distribute the 3 on the right-hand side:
[tex]\[ 6w - 15 \leq 3(-1) + 3w \][/tex]
[tex]\[ 6w - 15 \leq -3 + 3w \][/tex]
2. Subtract 3w from both sides to isolate the variable term on one side:
[tex]\[ 6w - 3w - 15 \leq -3 + 3w - 3w \][/tex]
[tex]\[ 3w - 15 \leq -3 \][/tex]
3. Add 15 to both sides:
[tex]\[ 3w - 15 + 15 \leq -3 + 15 \][/tex]
[tex]\[ 3w \leq 12 \][/tex]
4. Divide both sides by 3:
[tex]\[ w \leq 4 \][/tex]
We now have two inequalities that define the possible values for [tex]\( w \)[/tex]:
1. [tex]\( w > -1 \)[/tex]
2. [tex]\( w \leq 4 \)[/tex]
Combining these two inequalities, the solution set is:
[tex]\[ -1 < w \leq 4 \][/tex]
Let's check which of the given options fall within this range:
A) [tex]\( w = -1 \)[/tex]
- [tex]\(-1\)[/tex] is not greater than [tex]\(-1\)[/tex], so it does not satisfy [tex]\( -1 < w \)[/tex].
B) [tex]\( w = 2 \)[/tex]
- [tex]\( 2 \)[/tex] is greater than [tex]\(-1\)[/tex] and less than or equal to [tex]\( 4 \)[/tex], so it satisfies [tex]\( -1 < w \leq 4 \)[/tex].
C) [tex]\( w = 5 \)[/tex]
- [tex]\( 5 \)[/tex] is greater than [tex]\( 4 \)[/tex], so it does not satisfy [tex]\( w \leq 4 \)[/tex].
D) [tex]\( w = 8 \)[/tex]
- [tex]\( 8 \)[/tex] is greater than [tex]\( 4 \)[/tex], so it does not satisfy [tex]\( w \leq 4 \)[/tex].
Therefore, the number that satisfies both inequalities is:
[tex]\[ \boxed{2} \][/tex]
