To solve the inequalities [tex]\(5x - 4 \geq 12\)[/tex] and [tex]\(12x + 5 \leq -4\)[/tex], let’s break it down step-by-step.
### Solving the First Inequality: [tex]\(5x - 4 \geq 12\)[/tex]
1. Add 4 to both sides to isolate the term with [tex]\(x\)[/tex]:
[tex]\[
5x - 4 + 4 \geq 12 + 4
\][/tex]
Simplifying, we get:
[tex]\[
5x \geq 16
\][/tex]
2. Divide both sides by 5 to solve for [tex]\(x\)[/tex]:
[tex]\[
x \geq \frac{16}{5}
\][/tex]
Therefore,
[tex]\[
x \geq 3.2
\][/tex]
### Solving the Second Inequality: [tex]\(12x + 5 \leq -4\)[/tex]
1. Subtract 5 from both sides to isolate the term with [tex]\(x\)[/tex]:
[tex]\[
12x + 5 - 5 \leq -4 - 5
\][/tex]
Simplifying, we get:
[tex]\[
12x \leq -9
\][/tex]
2. Divide both sides by 12 to solve for [tex]\(x\)[/tex]:
[tex]\[
x \leq \frac{-9}{12}
\][/tex]
Simplifying the fraction, we get:
[tex]\[
x \leq -0.75
\][/tex]
### Conclusion
The solution to the inequalities is:
[tex]\[
x \geq 3.2 \quad \text{or} \quad x \leq -0.75
\][/tex]
This means that [tex]\(x\)[/tex] can be any value greater than or equal to 3.2, or any value less than or equal to -0.75.
