Answer :

GinnyAnswer
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].

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