Let's solve each inequality step-by-step to match it with its solution on the number line.
1. Inequality: [tex]\( x - 99 \leq -104 \)[/tex]
Step-by-step solution:
[tex]\[
x - 99 \leq -104
\][/tex]
Add [tex]\(99\)[/tex] to both sides:
[tex]\[
x \leq -104 + 99
\][/tex]
Simplify:
[tex]\[
x \leq -5
\][/tex]
Solution on the number line:
[tex]\[
(-\infty, -5]
\][/tex]
2. Inequality: [tex]\( x - 51 \leq -43 \)[/tex]
Step-by-step solution:
[tex]\[
x - 51 \leq -43
\][/tex]
Add [tex]\(51\)[/tex] to both sides:
[tex]\[
x \leq -43 + 51
\][/tex]
Simplify:
[tex]\[
x \leq 8
\][/tex]
Solution on the number line:
[tex]\[
(-\infty, 8]
\][/tex]
3. Inequality: [tex]\( 150 + x \leq 144 \)[/tex]
Step-by-step solution:
[tex]\[
150 + x \leq 144
\][/tex]
Subtract [tex]\(150\)[/tex] from both sides:
[tex]\[
x \leq 144 - 150
\][/tex]
Simplify:
[tex]\[
x \leq -6
\][/tex]
Solution on the number line:
[tex]\[
(-\infty, -6]
\][/tex]
4. Inequality: [tex]\( 75 < 69 - x \)[/tex]
Step-by-step solution:
[tex]\[
75 < 69 - x
\][/tex]
Subtract [tex]\(69\)[/tex] from both sides:
[tex]\[
75 - 69 < -x
\][/tex]
Simplify:
[tex]\[
6 < -x
\][/tex]
Divide both sides by [tex]\(-1\)[/tex] (remember to reverse inequality):
[tex]\[
-6 > x \implies x < -6
\][/tex]
Solution on the number line:
[tex]\[
(-\infty, -6)
\][/tex]
5. Matching each inequality to its solution range on the number line:
- For [tex]\( x - 99 \leq -104 \)[/tex], the solution is [tex]\( (-\infty, -5] \)[/tex];
- For [tex]\( x - 51 \leq -43 \)[/tex], the solution is [tex]\( (-\infty, 8] \)[/tex];
- For [tex]\( 150 + x \leq 144 \)[/tex], the solution is [tex]\( (-\infty, -6] \)[/tex];
- For [tex]\( 75 < 69 - x \)[/tex], the solution is [tex]\( (-\infty, -6) \)[/tex].
