To solve the inequality [tex]\(3 - 2x \leq 8\)[/tex], follow these steps:
1. Isolate the variable term:
[tex]\[
3 - 2x \leq 8
\][/tex]
Subtract 3 from both sides of the inequality:
[tex]\[
-2x \leq 8 - 3
\][/tex]
Simplify the right-hand side:
[tex]\[
-2x \leq 5
\][/tex]
2. Solve for [tex]\(x\)[/tex]:
Since we are dividing by a negative number to isolate [tex]\(x\)[/tex], we must reverse the inequality sign:
[tex]\[
x \geq \frac{5}{-2}
\][/tex]
Simplify the fraction:
[tex]\[
x \geq -\frac{5}{2}
\][/tex]
So the solution to the inequality [tex]\(3 - 2x \leq 8\)[/tex] is:
[tex]\[
x \geq -\frac{5}{2}
\][/tex]
3. Express the solution in interval notation:
[tex]\[
\left[ -\frac{5}{2}, \infty \right)
\][/tex]
4. Graph the solution on a number line:
- Draw a number line.
- Mark the point [tex]\(-\frac{5}{2}\)[/tex] (or -2.5) on the number line.
- Use a solid dot at [tex]\(-\frac{5}{2}\)[/tex] to indicate that it is included in the solution (since the inequality is [tex]\(\leq\)[/tex]).
- Shade the region to the right of [tex]\(-\frac{5}{2}\)[/tex] to represent all values greater than or equal to [tex]\(-\frac{5}{2}\)[/tex].
Here's how the graph should look:
```plaintext
<---(====)=================================================>
-3 -2.5 ∞
(====]: solid dot at -2.5 indicating -2.5 is included,
======>: arrow to indicate all numbers greater than or equal to -2.5.
```
This graph clearly shows that the solution to the inequality [tex]\(3 - 2x \leq 8\)[/tex] includes all values [tex]\(x \geq -\frac{5}{2}\)[/tex].
