Certainly! Let's solve the given inequalities step by step:
1. Solve the first inequality: [tex]\( 4x - 18 > 2 \)[/tex]
- Step 1: Add 18 to both sides to isolate the term with [tex]\( x \)[/tex]:
[tex]\[
4x - 18 + 18 > 2 + 18
\][/tex]
Simplifying this gives:
[tex]\[
4x > 20
\][/tex]
- Step 2: Divide both sides by 4 to solve for [tex]\( x \)[/tex]:
[tex]\[
\frac{4x}{4} > \frac{20}{4}
\][/tex]
Simplifying this gives:
[tex]\[
x > 5
\][/tex]
2. Solve the second inequality: [tex]\( -17x - 8 \leq -25 \)[/tex]
- Step 1: Add 8 to both sides to isolate the term with [tex]\( x \)[/tex]:
[tex]\[
-17x - 8 + 8 \leq -25 + 8
\][/tex]
Simplifying this gives:
[tex]\[
-17x \leq -17
\][/tex]
- Step 2: Divide both sides by -17. Note that dividing by a negative number reverses the inequality sign:
[tex]\[
\frac{-17x}{-17} \geq \frac{-17}{-17}
\][/tex]
Simplifying this gives:
[tex]\[
x \geq 1
\][/tex]
Now, combine the two conditions:
- From the first inequality: [tex]\( x > 5 \)[/tex]
- From the second inequality: [tex]\( x \geq 1 \)[/tex]
Both conditions must be satisfied simultaneously. The range that satisfies both is the stricter one, which is:
[tex]\[
x > 5
\][/tex]
So the solution to the problem is:
All real numbers [tex]\( x \)[/tex] such that [tex]\( x > 5 \)[/tex].
