Let's solve the compound inequality step-by-step:
### Solving the Inequalities Separately
1. First Inequality: [tex]\(-2 < 2x - 6\)[/tex]
- Step 1: To isolate [tex]\(x\)[/tex], add 6 to both sides of the inequality:
[tex]\[
-2 + 6 < 2x - 6 + 6
\][/tex]
Simplify:
[tex]\[
4 < 2x
\][/tex]
- Step 2: Divide both sides by 2:
[tex]\[
\frac{4}{2} < \frac{2x}{2}
\][/tex]
Simplify:
[tex]\[
2 < x
\][/tex]
This can also be written as:
[tex]\[
x > 2
\][/tex]
2. Second Inequality: [tex]\(3x - 12 \leq 10\)[/tex]
- Step 1: To isolate [tex]\(x\)[/tex], add 12 to both sides of the inequality:
[tex]\[
3x - 12 + 12 \leq 10 + 12
\][/tex]
Simplify:
[tex]\[
3x \leq 22
\][/tex]
- Step 2: Divide both sides by 3:
[tex]\[
\frac{3x}{3} \leq \frac{22}{3}
\][/tex]
Simplify:
[tex]\[
x \leq \frac{22}{3}
\][/tex]
Converting the fraction to a decimal:
[tex]\[
\frac{22}{3} \approx 7.333333333333333
\][/tex]
This is approximately:
[tex]\[
x \leq 7.333333333333333
\][/tex]
### Combining the Inequalities
To find the solution to the compound inequality [tex]\((-2 < 2x - 6) \cap (3x - 12 \leq 10)\)[/tex], we combine the two results:
[tex]\[
x > 2 \quad \text{and} \quad x \leq 7.333333333333333
\][/tex]
Graphically, this represents all the numbers [tex]\(x\)[/tex] that satisfy both conditions simultaneously. Therefore, the solution set is:
[tex]\[
2 < x \leq 7.333333333333333
\][/tex]
### Graphing the Solution
On a number line, we represent this as:
- An open circle at [tex]\(x = 2\)[/tex] (since [tex]\(x\)[/tex] must be greater than 2, but not equal to 2),
- A closed circle at [tex]\(x \approx 7.333333333333333\)[/tex] (since [tex]\(x\)[/tex] can be equal to [tex]\(7.333333333333333\)[/tex]),
- A shaded region between these two points indicating all [tex]\(x\)[/tex] such that [tex]\(x\)[/tex] is greater than 2 and less than or equal to [tex]\(7.333333333333333\)[/tex].
So, the graph of the compound inequality [tex]\((-2 < 2 x-6) \cap (3 x-12 \leq 10)\)[/tex] is represented as follows:
[tex]\[
\begin{array}{ccccccccccccccc}
& & 2 & & & & & & 7.333 \\
& (&-------&|--------)&
\end{array}
\][/tex]
