Certainly! Let's solve each inequality step by step and match them to the number line that represents their solution.
### Inequality 1: [tex]\( x - 99 \leq -104 \)[/tex]
1. Add 99 to both sides of the inequality:
[tex]\[
x - 99 + 99 \leq -104 + 99
\][/tex]
[tex]\[
x \leq -5
\][/tex]
The solution [tex]\( x \leq -5 \)[/tex] represents all values of [tex]\( x \)[/tex] that are less than or equal to [tex]\(-5\)[/tex].
### Inequality 2: [tex]\( x - 51 \leq -43 \)[/tex]
1. Add 51 to both sides of the inequality:
[tex]\[
x - 51 + 51 \leq -43 + 51
\][/tex]
[tex]\[
x \leq 8
\][/tex]
The solution [tex]\( x \leq 8 \)[/tex] represents all values of [tex]\( x \)[/tex] that are less than or equal to [tex]\(8\)[/tex].
### Inequality 3: [tex]\( 150 + x \leq 144 \)[/tex]
1. Subtract 150 from both sides of the inequality:
[tex]\[
150 + x - 150 \leq 144 - 150
\][/tex]
[tex]\[
x \leq -6
\][/tex]
The solution [tex]\( x \leq -6 \)[/tex] represents all values of [tex]\( x \)[/tex] that are less than or equal to [tex]\(-6\)[/tex].
### Inequality 4: [tex]\( 75 < 69 - x \)[/tex]
1. Add [tex]\( x \)[/tex] to both sides of the inequality (to isolate [tex]\( x \)[/tex] on one side):
[tex]\[
75 + x < 69
\][/tex]
2. Subtract 75 from both sides:
[tex]\[
x < 69 - 75
\][/tex]
[tex]\[
x < -6
\][/tex]
The solution [tex]\( x < -6 \)[/tex] represents all values of [tex]\( x \)[/tex] that are strictly less than [tex]\(-6\)[/tex].
### Summary and Matching to the Number Line
Given the solutions to each inequality:
- [tex]\( x \leq -5 \)[/tex]
- [tex]\( x \leq 8 \)[/tex]
- [tex]\( x \leq -6 \)[/tex]
- [tex]\( x < -6 \)[/tex]
We can match each inequality to its respective solution on the number line:
1. [tex]\( x - 99 \leq -104 \)[/tex]: This corresponds to [tex]\( x \leq -5 \)[/tex].
2. [tex]\( x - 51 \leq -43 \)[/tex]: This corresponds to [tex]\( x \leq 8 \)[/tex].
3. [tex]\( 150 + x \leq 144 \)[/tex]: This corresponds to [tex]\( x \leq -6 \)[/tex].
4. [tex]\( 75 < 69 - x \)[/tex]: This corresponds to [tex]\( x < -6 \)[/tex].
Thus, the inequalities and their solutions on the number line can be represented as follows:
[tex]\[
\begin{aligned}
& x - 99 \leq -104 &\quad \text{corresponds to} \quad x \leq -5 \\
& x - 51 \leq -43 &\quad \text{corresponds to} \quad x \leq 8 \\
& 150 + x \leq 144 &\quad \text{corresponds to} \quad x \leq -6 \\
& 75 < 69 - x &\quad \text{corresponds to} \quad x < -6 \\
\end{aligned}
\][/tex]
