To graph the solution set to the inequality [tex]\(-2x + 9 \leq 5x - 12\)[/tex], we first need to solve the inequality step-by-step. Let's break it down:
1. Combine like terms:
[tex]\[
-2x + 9 \leq 5x - 12
\][/tex]
Subtract [tex]\(5x\)[/tex] from both sides to move all the [tex]\(x\)[/tex]-terms to one side of the inequality:
[tex]\[
-2x - 5x + 9 \leq -12
\][/tex]
Simplify:
[tex]\[
-7x + 9 \leq -12
\][/tex]
2. Isolate the [tex]\(x\)[/tex]-term:
Subtract 9 from both sides:
[tex]\[
-7x + 9 - 9 \leq -12 - 9
\][/tex]
Simplify:
[tex]\[
-7x \leq -21
\][/tex]
3. Solve for [tex]\(x\)[/tex]:
Divide both sides by [tex]\(-7\)[/tex]. Remember, when you divide by a negative number, you need to flip the inequality sign:
[tex]\[
x \geq 3
\][/tex]
Now we have the solution [tex]\(x \geq 3\)[/tex].
### Graphing the Solution Set:
To graph the solution on a number line:
1. Draw a number line:
Place a number line with values marked at regular intervals (usually integers).
2. Locate the point [tex]\(3\)[/tex]:
Identify the position of [tex]\(3\)[/tex] on the number line.
3. Use an open circle or closed circle:
- Since the inequality is [tex]\(\geq\)[/tex] (greater than or equal to), place a closed circle at [tex]\(3\)[/tex]. A closed circle indicates that the point [tex]\(3\)[/tex] is included in the solution set.
4. Draw the ray:
- Draw a ray starting at the point [tex]\(3\)[/tex] and extending to the right, indicating all values greater than or equal to [tex]\(3\)[/tex].
Here is a visual representation on the number line:
[tex]\[
\begin{array}{cccccccccccccccc}
\ & \ & \ & \ & \ & \ & \ & \ & \bullet & \rightarrow \\
\ & -2 & \ & -1 & \ & 0 & \ & 1 & \ & 2 & \ & 3 & \ & 4 & \ & 5
\end{array}
\][/tex]
- The closed circle at [tex]\(3\)[/tex] shows that [tex]\(3\)[/tex] is included in the solution set.
- The ray to the right of [tex]\(3\)[/tex] indicates all values [tex]\(x\)[/tex] such that [tex]\(x \geq 3\)[/tex].
