Sure! Let's solve the given inequality step by step and then graph the solution set on the number line.
### Step 1: Write the inequality
[tex]\[
-2x + 9 \leq 5x - 12
\][/tex]
### Step 2: Isolate the variable term on one side
First, add [tex]\(2x\)[/tex] to both sides to move all [tex]\(x\)[/tex]-terms to one side of the inequality:
[tex]\[
9 \leq 7x - 12
\][/tex]
### Step 3: Isolate the constant term
Next, add 12 to both sides to move the constants to one side:
[tex]\[
21 \leq 7x
\][/tex]
### Step 4: Solve for [tex]\(x\)[/tex]
Now, divide both sides by 7:
[tex]\[
3 \leq x
\][/tex]
or equivalently
[tex]\[
x \geq 3
\][/tex]
### Step 5: Graph the solution on the number line
The solution set is [tex]\(x \geq 3\)[/tex]. To graph this inequality on a number line:
1. Draw a number line with a point at [tex]\(3\)[/tex].
2. Use a closed circle at [tex]\(3\)[/tex] to indicate that [tex]\(3\)[/tex] is included in the solution set (because [tex]\(3\)[/tex] satisfies the inequality [tex]\(3 \leq x\)[/tex]).
3. Shade the number line to the right of [tex]\(3\)[/tex], indicating that all numbers greater than or equal to [tex]\(3\)[/tex] are part of the solution set.
Here is the graphical representation:
[tex]\[
\begin{array}{c}
\begin{tikzpicture}
\draw[thick, ->] (-1,0) -- (6,0) node[anchor=north west] {x};
\foreach \x in {0,1,2,3,4,5}
\draw (\x,0.1) -- (\x,-0.1) node[anchor=north] {\x};
\fill (3,0) circle (3pt);
\draw[thick, ->] (3,0) -- (6,0);
\end{tikzpicture}
\end{array}
\][/tex]
This number line indicates that the solution set for the inequality [tex]\(-2x + 9 \leq 5x - 12\)[/tex] is [tex]\(x \geq 3\)[/tex].
