Answer :
To solve the compound inequality [tex]\(5x - 4 \geq 12 \quad \text{OR} \quad 12x + 5 \leq -4\)[/tex], we need to solve each inequality separately and then combine the solutions.
### 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]
[tex]\[ 5x \geq 16 \][/tex]
2. Divide both sides by 5:
[tex]\[ x \geq \frac{16}{5} \][/tex]
[tex]\[ x \geq 3.2 \][/tex]
The solution to the first inequality is [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]
[tex]\[ 12x \leq -9 \][/tex]
2. Divide both sides by 12:
[tex]\[ x \leq \frac{-9}{12} \][/tex]
Simplify the fraction:
[tex]\[ x \leq -\frac{3}{4} \][/tex]
[tex]\[ x \leq -0.75 \][/tex]
The solution to the second inequality is [tex]\(x \leq -0.75\)[/tex].
### Combining the Solutions
Since the original problem asks for [tex]\(x\)[/tex] values that satisfy either inequality, we combine the two sets of solutions. Therefore, the solution to the compound inequality [tex]\(5x - 4 \geq 12 \quad \text{OR} \quad 12x + 5 \leq -4\)[/tex] is:
[tex]\[ x \geq 3.2 \quad \text{OR} \quad x \leq -0.75 \][/tex]
In interval notation, this solution can be expressed as:
[tex]\[ (-\infty, -0.75] \cup [3.2, \infty) \][/tex]
So, the final answer is:
[tex]\[ (-\infty, -0.75] \cup [3.2, \infty) \][/tex]
### 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]
[tex]\[ 5x \geq 16 \][/tex]
2. Divide both sides by 5:
[tex]\[ x \geq \frac{16}{5} \][/tex]
[tex]\[ x \geq 3.2 \][/tex]
The solution to the first inequality is [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]
[tex]\[ 12x \leq -9 \][/tex]
2. Divide both sides by 12:
[tex]\[ x \leq \frac{-9}{12} \][/tex]
Simplify the fraction:
[tex]\[ x \leq -\frac{3}{4} \][/tex]
[tex]\[ x \leq -0.75 \][/tex]
The solution to the second inequality is [tex]\(x \leq -0.75\)[/tex].
### Combining the Solutions
Since the original problem asks for [tex]\(x\)[/tex] values that satisfy either inequality, we combine the two sets of solutions. Therefore, the solution to the compound inequality [tex]\(5x - 4 \geq 12 \quad \text{OR} \quad 12x + 5 \leq -4\)[/tex] is:
[tex]\[ x \geq 3.2 \quad \text{OR} \quad x \leq -0.75 \][/tex]
In interval notation, this solution can be expressed as:
[tex]\[ (-\infty, -0.75] \cup [3.2, \infty) \][/tex]
So, the final answer is:
[tex]\[ (-\infty, -0.75] \cup [3.2, \infty) \][/tex]
