Answer :

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.

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