To solve the system of inequalities, we need to address each inequality separately and then find the common solution. Let's break this down step-by-step:
1. Solve the first inequality: [tex]\( 17x + 5 > 39 \)[/tex]
- Subtract 5 from both sides to isolate the term with [tex]\( x \)[/tex]:
[tex]\[
17x + 5 - 5 > 39 - 5
\][/tex]
[tex]\[
17x > 34
\][/tex]
- Divide both sides by 17 to solve for [tex]\( x \)[/tex]:
[tex]\[
x > \frac{34}{17}
\][/tex]
[tex]\[
x > 2
\][/tex]
So, the solution to the first inequality is:
[tex]\[
x > 2
\][/tex]
2. Solve the second inequality: [tex]\( -13x - 6 > -45 \)[/tex]
- Add 6 to both sides to isolate the term with [tex]\( x \)[/tex]:
[tex]\[
-13x - 6 + 6 > -45 + 6
\][/tex]
[tex]\[
-13x > -39
\][/tex]
- Divide both sides by -13. Remember, when dividing by a negative number, the inequality sign reverses:
[tex]\[
x < \frac{-39}{-13}
\][/tex]
[tex]\[
x < 3
\][/tex]
So, the solution to the second inequality is:
[tex]\[
x < 3
\][/tex]
3. Combine the solutions:
We have:
[tex]\[
x > 2
\][/tex]
and
[tex]\[
x < 3
\][/tex]
The common solution is the intersection of these two intervals. Therefore:
[tex]\[
2 < x < 3
\][/tex]
So, the final solution to the system of inequalities [tex]\( 17x + 5 > 39 \)[/tex] and [tex]\( -13x - 6 > -45 \)[/tex] is:
[tex]\[
2 < x < 3
\][/tex]
