Sure, let's solve the inequality [tex]\(5 < 5x \leq 20\)[/tex] step by step.
1. Start with the given inequality:
[tex]\[
5 < 5x \leq 20
\][/tex]
2. Split the compound inequality into two separate inequalities:
[tex]\[
5 < 5x \quad \text{and} \quad 5x \leq 20
\][/tex]
3. Solve the first inequality [tex]\(5 < 5x\)[/tex]:
- Divide both sides by 5 to isolate [tex]\(x\)[/tex]:
[tex]\[
\frac{5}{5} < \frac{5x}{5}
\][/tex]
[tex]\[
1 < x
\][/tex]
- This simplifies to:
[tex]\[
x > 1
\][/tex]
4. Solve the second inequality [tex]\(5x \leq 20\)[/tex]:
- Again, divide both sides by 5 to isolate [tex]\(x\)[/tex]:
[tex]\[
\frac{5x}{5} \leq \frac{20}{5}
\][/tex]
[tex]\[
x \leq 4
\][/tex]
5. Combine the two results from steps 3 and 4:
- From [tex]\(5 < 5x\)[/tex], we have [tex]\(x > 1\)[/tex].
- From [tex]\(5x \leq 20\)[/tex], we have [tex]\(x \leq 4\)[/tex].
Combining these two,
[tex]\[
1 < x \leq 4
\][/tex]
6. Conclusion:
The solution set for the inequality [tex]\(5 < 5x \leq 20\)[/tex] is:
[tex]\[
(1, 4]
\][/tex]
This means that [tex]\(x\)[/tex] is any number greater than 1 and up to and including 4.
