Certainly! Let's break down and solve the given inequalities step by step.
### Inequality 1: [tex]\(-x < 9 - 12\)[/tex]
1. Simplify the right-hand side:
[tex]\[
9 - 12 = -3
\][/tex]
So the inequality becomes:
[tex]\[
-x < -3
\][/tex]
2. To isolate [tex]\(x\)[/tex], multiply both sides of the inequality by [tex]\(-1\)[/tex]. Remember that multiplying or dividing an inequality by a negative number reverses the inequality sign:
[tex]\[
x > 3
\][/tex]
### Inequality 2: [tex]\(-20 < -x - 9\)[/tex]
1. Add 9 to both sides to isolate [tex]\(-x\)[/tex]:
[tex]\[
-20 + 9 < -x
\][/tex]
Simplify the left-hand side:
[tex]\[
-11 < -x
\][/tex]
2. Again, multiply both sides by [tex]\(-1\)[/tex], reversing the inequality sign:
[tex]\[
11 > x
\][/tex]
This can also be written as:
[tex]\[
x < 11
\][/tex]
### Combining the Solutions
We have two inequalities:
1. [tex]\(x > 3\)[/tex]
2. [tex]\(x < 11\)[/tex]
When we combine these inequalities, we find that [tex]\(x\)[/tex] must satisfy both conditions simultaneously. This gives us the combined inequality:
[tex]\[
3 < x < 11
\][/tex]
### Graphing the Solution
To graph the solution [tex]\(3 < x < 11\)[/tex] on a number line:
1. Draw a number line.
2. Mark the numbers 3 and 11 on the number line.
3. Place open circles at 3 and 11 to indicate that these points are not included in the solution (since the inequality is strict, [tex]\(>\)[/tex] and [tex]\(<\)[/tex] rather than [tex]\(\geq\)[/tex] and [tex]\(\leq\)[/tex]).
4. Shade the region between 3 and 11 to indicate that all numbers in this interval are solutions.
The graph on the number line should look like this:
```
2 3 4 5 6 7 8 9 10 11 12
|---(===)===================(==)---|
```
The open circles at 3 and 11 indicate that these endpoints are not part of the solution, and the shaded region shows all the values of [tex]\(x\)[/tex] that satisfy [tex]\(3 < x < 11\)[/tex].
