Sure! Let's solve the inequality step-by-step and then graph the solution on the number line.
1. Start with the given inequality:
[tex]\[
-2x + 9 \leq 5x - 12
\][/tex]
2. Isolate the variable [tex]\( x \)[/tex] on one side of the inequality. Let's move [tex]\( -2x \)[/tex] to the right-hand side by adding [tex]\( 2x \)[/tex] to both sides:
[tex]\[
9 \leq 5x + 2x - 12
\][/tex]
3. Simplify the inequality:
[tex]\[
9 \leq 7x - 12
\][/tex]
4. Next, add 12 to both sides to isolate terms involving [tex]\( x \)[/tex]:
[tex]\[
9 + 12 \leq 7x
\][/tex]
5. Simplify again:
[tex]\[
21 \leq 7x
\][/tex]
6. Finally, divide both sides by 7 to solve for [tex]\( x \)[/tex]:
[tex]\[
3 \leq x
\][/tex]
We can rewrite this as:
[tex]\[
x \geq 3
\][/tex]
Now, let's graph the solution [tex]\( x \geq 3 \)[/tex] on the number line.
1. Draw a number line.
2. Locate the critical point [tex]\( x = 3 \)[/tex] and mark it on the number line.
3. Since the inequality includes [tex]\( x = 3 \)[/tex], represent this with a closed (solid) dot at 3.
4. Shade the region to the right of 3 to indicate all values greater than or equal to 3.
Here is the graph on the number line:
[tex]\[
\begin{array}{ccccccccccccccccccc}
& & & & & & & & & \bullet & \xrightarrow{\hspace{3cm}} & & & & & & & & \\
& - & 2 & & - & 1 & & 0 & & 1 & & 2 & & 3 & & 4 & & 5 & & \\
\end{array}
\][/tex]
The solid dot at 3 and the shading to the right indicate that [tex]\( x \geq 3 \)[/tex] is the solution to the inequality.
