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Solve the inequality:

[tex]7 < x + 3 \leq 15[/tex]

Answer :

To solve the inequality [tex]\(7 < x + 3 \leq 15\)[/tex], we can break it into two separate inequalities and solve each one step by step.

1. Solve [tex]\(7 < x + 3\)[/tex]:

- Subtract 3 from both sides to isolate [tex]\(x\)[/tex]:
[tex]\[
7 - 3 < x
\][/tex]
[tex]\[
4 < x
\][/tex]
- This can be rewritten as:
[tex]\[
x > 4
\][/tex]

2. Solve [tex]\(x + 3 \leq 15\)[/tex]:

- Subtract 3 from both sides to isolate [tex]\(x\)[/tex]:
[tex]\[
x + 3 - 3 \leq 15 - 3
\][/tex]
[tex]\[
x \leq 12
\][/tex]

3. Combine the solutions:

- Now, combine the results from both parts:
[tex]\[
4 < x \leq 12
\][/tex]

So, the solution to the inequality [tex]\(7 < x + 3 \leq 15\)[/tex] is that [tex]\(x\)[/tex] is in the range [tex]\(4 < x \leq 12\)[/tex]. In interval notation, this solution is expressed as [tex]\((4, 12]\)[/tex].

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