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].
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].
