Sure! Let's break down the problem and solve the inequalities step-by-step.
### Solving the first inequality:
[tex]\[ 4x - 18 > 2 \][/tex]
1. Add 18 to both sides of the inequality:
[tex]\[ 4x - 18 + 18 > 2 + 18 \][/tex]
[tex]\[ 4x > 20 \][/tex]
2. Divide both sides by 4:
[tex]\[ \frac{4x}{4} > \frac{20}{4} \][/tex]
[tex]\[ x > 5 \][/tex]
### Solving the second inequality:
[tex]\[ -17x - 8 \geq -25 \][/tex]
1. Add 8 to both sides of the inequality:
[tex]\[ -17x - 8 + 8 \geq -25 + 8 \][/tex]
[tex]\[ -17x \geq -17 \][/tex]
2. Divide both sides by -17, and remember to reverse the inequality when dividing by a negative number:
[tex]\[ \frac{-17x}{-17} \leq \frac{-17}{-17} \][/tex]
[tex]\[ x \leq 1 \][/tex]
### Combining the solutions:
The original problem involves the logical "or" operator, so we need to find all [tex]\(x\)[/tex] that satisfy either of the inequalities:
[tex]\[ x > 5 \quad \text{or} \quad x \leq 1 \][/tex]
### Interval Notation:
1. [tex]\( x > 5 \)[/tex] is the interval [tex]\( (5, \infty) \)[/tex]
2. [tex]\( x \leq 1 \)[/tex] is the interval [tex]\( (-\infty, 1] \)[/tex]
Thus, the solution set combining both intervals is:
[tex]\[ (-\infty, 1] \cup (5, \infty) \][/tex]
So, the set of all real numbers [tex]\(x\)[/tex] that satisfy the given inequalities is:
[tex]\[ \boxed{(-\infty, 1] \cup (5, \infty)} \][/tex]
