To solve the inequalities [tex]\( -10 < -x - 9 \)[/tex] and [tex]\( -x - 9 \geq -15 \)[/tex], let's tackle each inequality step-by-step.
### First Inequality: [tex]\( -10 < -x - 9 \)[/tex]
1. Add 9 to both sides to isolate the term with [tex]\( x \)[/tex]:
[tex]\[
-10 + 9 < -x - 9 + 9
\][/tex]
2. Simplify the inequality:
[tex]\[
-1 < -x
\][/tex]
3. Multiply both sides by [tex]\(-1\)[/tex] to solve for [tex]\( x \)[/tex], remembering to reverse the inequality:
[tex]\[
1 > x
\][/tex]
This can also be written as:
[tex]\[
x < 1
\][/tex]
### Second Inequality: [tex]\( -x - 9 \geq -15 \)[/tex]
1. Add 9 to both sides to isolate the term with [tex]\( x \)[/tex]:
[tex]\[
-x - 9 + 9 \geq -15 + 9
\][/tex]
2. Simplify the inequality:
[tex]\[
-x \geq -6
\][/tex]
3. Multiply both sides by [tex]\(-1\)[/tex] to solve for [tex]\( x \)[/tex], remembering to reverse the inequality:
[tex]\[
x \leq 6
\][/tex]
### Combining the Results
Now we combine the results of both inequalities:
- From the first inequality: [tex]\( x < 1 \)[/tex]
- From the second inequality: [tex]\( x \leq 6 \)[/tex]
The solution is all values of [tex]\( x \)[/tex] that satisfy both conditions:
[tex]\[
x < 1 \quad \text{and} \quad x \leq 6
\][/tex]
Since [tex]\( x \)[/tex] must satisfy [tex]\( x < 1 \)[/tex], which is a stricter condition than [tex]\( x \leq 6 \)[/tex], the combined solution is:
[tex]\[
x < 1
\][/tex]
Therefore, the final solution to the system of inequalities is:
[tex]\[
\boxed{x < 1}
\][/tex]
