Certainly! Let's solve each linear inequality step-by-step and then determine the combined solution on the number line.
### Inequality 1: [tex]\( 4 - 3x < 7 \)[/tex]
1. Subtract 4 from both sides:
[tex]\[
4 - 3x - 4 < 7 - 4
\][/tex]
[tex]\[
-3x < 3
\][/tex]
2. Divide both sides by -3 (note that dividing by a negative number reverses the inequality):
[tex]\[
x > -1
\][/tex]
So, the solution to the first inequality is:
[tex]\[
x > -1
\][/tex]
### Inequality 2: [tex]\( 8 - 6x \geq -16 \)[/tex]
1. Subtract 8 from both sides:
[tex]\[
8 - 6x - 8 \geq -16 - 8
\][/tex]
[tex]\[
-6x \geq -24
\][/tex]
2. Divide both sides by -6 (again, remember to reverse the inequality):
[tex]\[
x \leq 4
\][/tex]
So, the solution to the second inequality is:
[tex]\[
x \leq 4
\][/tex]
### Combined Solution
We now have two inequalities:
[tex]\[
x > -1
\][/tex]
[tex]\[
x \leq 4
\][/tex]
To find the combined solution, we need the values of [tex]\( x \)[/tex] that satisfy both inequalities simultaneously. Hence, we take the intersection of the two sets:
[tex]\[
\begin{cases}
x > -1 \\
x \leq 4
\end{cases}
\][/tex]
The combined solution is:
[tex]\[
-1 < x \leq 4
\][/tex]
### Representation on the Number Line
To represent the solution [tex]\(-1 < x \leq 4\)[/tex] on the number line:
- Use an open circle at [tex]\( x = -1 \)[/tex] to indicate that [tex]\(-1\)[/tex] is not included.
- Use a closed circle at [tex]\( x = 4 \)[/tex] to indicate that [tex]\(4\)[/tex] is included.
- Shade the region between [tex]\(-1\)[/tex] and [tex]\(4\)[/tex].
```plaintext
Number line representation:
-2 -1 0 1 2 3 4 5
(------------------------]
```
The segment [tex]\((------------------------]\)[/tex] represents all the values of [tex]\( x \)[/tex] satisfying the inequalities, from just greater than [tex]\(-1\)[/tex] to [tex]\(4\)[/tex], inclusive.
