Let's solve each inequality step-by-step and find the solution for each.
### Inequality 1:
[tex]\[ x - 99 \leq -104 \][/tex]
To solve for [tex]\( x \)[/tex], add 99 to both sides:
[tex]\[ x - 99 + 99 \leq -104 + 99 \][/tex]
[tex]\[ x \leq -5 \][/tex]
### Inequality 2:
[tex]\[ x - 51 \leq -43 \][/tex]
To solve for [tex]\( x \)[/tex], add 51 to both sides:
[tex]\[ x - 51 + 51 \leq -43 + 51 \][/tex]
[tex]\[ x \leq 8 \][/tex]
### Inequality 3:
[tex]\[ 150 + x \leq 144 \][/tex]
To solve for [tex]\( x \)[/tex], subtract 150 from both sides:
[tex]\[ 150 + x - 150 \leq 144 - 150 \][/tex]
[tex]\[ x \leq -6 \][/tex]
### Inequality 4:
[tex]\[ 75 < 69 - x \][/tex]
First, subtract 69 from both sides:
[tex]\[ 75 - 69 < 69 - x - 69 \][/tex]
[tex]\[ 6 < -x \][/tex]
Multiplying both sides by -1 and reversing the inequality sign:
[tex]\[ -6 > x \][/tex]
Or equivalently:
[tex]\[ x < -6 \][/tex]
Now, match each inequality to its corresponding solution on the number line:
1. [tex]\( x - 99 \leq -104 \)[/tex] matches [tex]\( x \leq -5 \)[/tex]
2. [tex]\( x - 51 \leq -43 \)[/tex] matches [tex]\( x \leq 8 \)[/tex]
3. [tex]\( 150 + x \leq 144 \)[/tex] matches [tex]\( x \leq -6 \)[/tex]
4. [tex]\( 75 < 69 - x \)[/tex] matches [tex]\( x < -6 \)[/tex]
Therefore, the matches are:
- [tex]\( x - 99 \leq -104 \)[/tex] [tex]\(\rightarrow\)[/tex] [tex]\( x \leq -5 \)[/tex]
- [tex]\( x - 51 \leq -43 \)[/tex] [tex]\(\rightarrow\)[/tex] [tex]\( x \leq 8 \)[/tex]
- [tex]\( 150 + x \leq 144 \)[/tex] [tex]\(\rightarrow\)[/tex] [tex]\( x \leq -6 \)[/tex]
- [tex]\( 75 < 69 - x \)[/tex] [tex]\(\rightarrow\)[/tex] [tex]\( x < -6 \)[/tex]
These steps provide a detailed solution for matching each inequality with its respective range on the number line.
