Answer :
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.
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.
