To solve the compound inequality, we need to solve each inequality separately and then find the intersection of the solutions.
First inequality: [tex]\(35x - 10 > -3\)[/tex]
1. Add 10 to both sides to isolate the term with [tex]\(x\)[/tex]:
[tex]\[
35x - 10 + 10 > -3 + 10 \implies 35x > 7
\][/tex]
2. Divide both sides by 35 to solve for [tex]\(x\)[/tex]:
[tex]\[
x > \frac{7}{35} \implies x > \frac{1}{5}
\][/tex]
So, the first part of our solution is [tex]\(x > 0.2\)[/tex].
Second inequality: [tex]\(8x - 9 < 39\)[/tex]
1. Add 9 to both sides to isolate the term with [tex]\(x\)[/tex]:
[tex]\[
8x - 9 + 9 < 39 + 9 \implies 8x < 48
\][/tex]
2. Divide both sides by 8 to solve for [tex]\(x\)[/tex]:
[tex]\[
x < \frac{48}{8} \implies x < 6
\][/tex]
So, the second part of our solution is [tex]\(x < 6\)[/tex].
Now, we combine both parts of the solution:
[tex]\[
0.2 < x < 6
\][/tex]
Thus, the solution set for the compound inequality is [tex]\(0.2 < x < 6\)[/tex].
Looking at the given options:
- [tex]\(-2 < x < 3 \frac{3}{4}\)[/tex]
- [tex]\(-2 < x < 6\)[/tex]
- [tex]\(2 < x < 6\)[/tex]
- [tex]\(2 < x < 3 \frac{3}{4}\)[/tex]
The correct solution set that matches our result is:
[tex]\[
-2 < x < 6
\][/tex]
So, the correct answer is [tex]\(-2 < x < 6\)[/tex].
