Let's solve the inequality step-by-step to graph its solution on a number line:
Given:
[tex]\[
\frac{3}{7}(35 x - 14) \leq \frac{21 x}{2} + 3
\][/tex]
Step 1: Simplify the left side of the inequality.
[tex]\[
\frac{3}{7}(35 x - 14) = \frac{3}{7} \cdot 35 x - \frac{3}{7} \cdot 14 = 3(5 x) - 6 = 15 x - 6
\][/tex]
So the inequality becomes:
[tex]\[
15 x - 6 \leq \frac{21 x}{2} + 3
\][/tex]
Step 2: Clear the fraction on the right side by multiplying every term by 2:
[tex]\[
2(15 x - 6) \leq 2 \left(\frac{21 x}{2} + 3\right)
\][/tex]
[tex]\[
30 x - 12 \leq 21 x + 6
\][/tex]
Step 3: Move all the terms involving [tex]\(x\)[/tex] to one side and constant terms to the other side:
[tex]\[
30 x - 21 x \leq 6 + 12
\][/tex]
[tex]\[
9 x \leq 18
\][/tex]
Step 4: Solve for [tex]\(x\)[/tex]:
[tex]\[
x \leq 2
\][/tex]
This means the solution set for the inequality is all [tex]\(x\)[/tex] values that are less than or equal to 2.
Step 5: Graph the solution on the number line.
To graph this, you would place a closed dot (or filled-in circle) at [tex]\(x = 2\)[/tex] to indicate that 2 is included in the solution set, and shade the entire number line to the left of 2 to represent all values less than or equal to 2.
[tex]\[
\begin{array}{rcl}
\text{Number line} & : & \quad \bullet \longrightarrow\\
& & ←——————————— \bullet \rightarrow \\
-3 & -2 & -1 \quad 0 \quad 1 \quad 2 \quad 3 \quad 4 \quad 5\\
\end{array}
\][/tex]
That's the solution on the number line.
