Let's solve the given inequalities step-by-step.
1. First Inequality: [tex]\(0 < n - 1 + 6\)[/tex]
[tex]\[
0 < n - 1 + 6
\][/tex]
Simplify inside the inequality:
[tex]\[
0 < n + 5
\][/tex]
Subtract 5 from both sides to isolate [tex]\(n\)[/tex]:
[tex]\[
0 - 5 < n
\][/tex]
[tex]\[
-5 < n
\][/tex]
So, the solution for this inequality is:
[tex]\[
n > -5
\][/tex]
2. Second Inequality: [tex]\(0 \geq n - 1 + 7\)[/tex]
[tex]\[
0 \geq n - 1 + 7
\][/tex]
Simplify inside the inequality:
[tex]\[
0 \geq n + 6
\][/tex]
Subtract 6 from both sides to isolate [tex]\(n\)[/tex]:
[tex]\[
0 - 6 \geq n
\][/tex]
[tex]\[
-6 \geq n
\][/tex]
So, the solution for this inequality is:
[tex]\[
n \leq -6
\][/tex]
3. Third Inequality: [tex]\(-3x + 2x > 5\)[/tex]
[tex]\[
-3x + 2x > 5
\][/tex]
Combine like terms:
[tex]\[
-x > 5
\][/tex]
Multiply both sides by -1 (remember to flip the inequality sign):
[tex]\[
x < -5
\][/tex]
So, the solution for this inequality is:
[tex]\[
x < -5
\][/tex]
4. Fourth Inequality: [tex]\(-3x + 2x \leq 6\)[/tex]
[tex]\[
-3x + 2x \leq 6
\][/tex]
Combine like terms:
[tex]\[
-x \leq 6
\][/tex]
Multiply both sides by -1 (remember to flip the inequality sign):
[tex]\[
x \geq -6
\][/tex]
So, the solution for this inequality is:
[tex]\[
x \geq -6
\][/tex]
To summarize, the solutions to the inequalities are:
- [tex]\(n > -5\)[/tex]
- [tex]\(n \leq -6\)[/tex]
- [tex]\(x < -5\)[/tex]
- [tex]\(x \geq -6\)[/tex]
