Certainly! To graph the solution of the inequality
[tex]\[
\frac{3}{7}(35x - 14) \leq \frac{21x}{2} + 3
\][/tex]
we need to follow a series of steps to solve it step by step and then represent this solution on a number line.
### Step-by-Step Solution:
1. Start with the given inequality:
[tex]\[
\frac{3}{7}(35x - 14) \leq \frac{21x}{2} + 3
\][/tex]
2. Distribute the [tex]\( \frac{3}{7} \)[/tex] on the left side:
[tex]\[
\frac{3}{7} \cdot 35x - \frac{3}{7} \cdot 14 \leq \frac{21x}{2} + 3
\][/tex]
Simplifying inside the parentheses:
[tex]\[
\frac{3 \cdot 35x}{7} - \frac{3 \cdot 14}{7} \leq \frac{21x}{2} + 3
\][/tex]
This simplifies to:
[tex]\[
15x - 6 \leq \frac{21x}{2} + 3
\][/tex]
3. Clear the fraction by finding a common denominator. Multiply through by 2 to get rid of the denominator:
[tex]\[
2(15x - 6) \leq 2\left(\frac{21x}{2} + 3\right)
\][/tex]
Which simplifies to:
[tex]\[
30x - 12 \leq 21x + 6
\][/tex]
4. Isolate the variable [tex]\(x\)[/tex]. Subtract [tex]\(21x\)[/tex] from both sides to start:
[tex]\[
30x - 21x - 12 \leq 6
\][/tex]
Combining like terms:
[tex]\[
9x - 12 \leq 6
\][/tex]
5. Add 12 to both sides to further isolate [tex]\(x\)[/tex]:
[tex]\[
9x \leq 18
\][/tex]
6. Finally, divide both sides by 9:
[tex]\[
x \leq 2
\][/tex]
### Solution Representation:
The solution to the inequality is:
[tex]\[ x \leq 2 \][/tex]
### Graphing on a Number Line:
1. Draw a number line.
2. Mark the point [tex]\( x = 2 \)[/tex].
3. Shade the region to the left of [tex]\( x = 2 \)[/tex] to represent all values less than or equal to 2.
4. Include a solid circle (darkened or filled-in) at [tex]\( x = 2 \)[/tex] to show that 2 is included in the solution set.
Here's a sketch of the number line:
[tex]\[ \dots \quad \longrightarrow \quad \bullet \quad \smash{0}\quad \smash {1} \quad \smash {2} \quad \smash{3} \quad \longrightarrow \quad \dots \][/tex]
- The shaded region extends leftwards from 2.
- A solid circle at [tex]\( x = 2 \)[/tex].
This represents all values [tex]\( x \leq 2 \)[/tex].
