Solve for [tex]x[/tex].

[tex]\[ 7x - 9 \ \textless \ 12 \quad \text{or} \quad \frac{1}{4}x + 8 \ \textgreater \ 11 \][/tex]

Enter your answer, including the inequality symbol, in the boxes.

[tex]\[ x \ \square \quad \text{or} \quad x \ \square \][/tex]



Answer :

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]