To solve the system of inequalities, we need to consider each inequality separately and then combine their solutions.
1. Solving the first inequality:
[tex]\[
7x - 9 < 12
\][/tex]
Step-by-Step:
- Add 9 to both sides:
[tex]\[
7x - 9 + 9 < 12 + 9 \implies 7x < 21
\][/tex]
- Divide both sides by 7:
[tex]\[
\frac{7x}{7} < \frac{21}{7} \implies x < 3
\][/tex]
So, the solution to the first inequality is:
[tex]\[
x < 3
\][/tex]
2. Solving the second inequality:
[tex]\[
\frac{1}{4}x + 8 > 11
\][/tex]
Step-by-Step:
- Subtract 8 from both sides:
[tex]\[
\frac{1}{4}x + 8 - 8 > 11 - 8 \implies \frac{1}{4}x > 3
\][/tex]
- Multiply both sides by 4:
[tex]\[
4 \cdot \frac{1}{4}x > 4 \cdot 3 \implies x > 12
\][/tex]
So, the solution to the second inequality is:
[tex]\[
x > 12
\][/tex]
3. Combining the solutions:
The two inequalities are connected by an "or". Therefore, the final solution combines the solutions of the individual inequalities.
- For the first inequality:
[tex]\[
x < 3
\][/tex]
- For the second inequality:
[tex]\[
x > 12
\][/tex]
Thus, the combined solution is:
[tex]\[
x < 3 \text{ or } x > 12
\][/tex]
### Final Answer:
[tex]\[
x < 3 \text{ or } x > 12
\][/tex]
