The problem asks to match each of the six given inequalities to the graph that represents its solution set.

1. Problem Description

The problem asks to match each of the six given inequalities to the graph that represents its solution set.

2. Solution Steps

1. $6x \le 3x$

Subtract $3x$ from both sides:

$6x - 3x \le 0$

$3x \le 0$

$x \le 0$

This corresponds to graph F, a closed circle at 0 and an arrow pointing to the left.

2. $\frac{1}{4} x > -\frac{1}{2}$

Multiply both sides by 4:

$x > -2$

This corresponds to graph A, an open circle at -2 and an arrow pointing to the right.

3. $5x + 4 \ge 7x$

Subtract $5x$ from both sides:

$4 \ge 2x$

Divide by 2:

$2 \ge x$ or $x \le 2$

This corresponds to graph C, a closed circle at 2 and an arrow pointing to the left.

4. $8x - 2 < -4(x - 1)$

$8x - 2 < -4x + 4$

Add $4x$ to both sides:

$12x - 2 < 4$

Add 2 to both sides:

$12x < 6$

Divide by 12:

$x < \frac{6}{12}$

$x < \frac{1}{2}$

This corresponds to graph D, an open circle at 1/2 and an arrow pointing to the left.

5. $\frac{4x - 1}{3} > -1$

Multiply both sides by 3:

$4x - 1 > -3$

Add 1 to both sides:

$4x > -2$

Divide by 4:

$x > -\frac{2}{4}$

$x > -\frac{1}{2}$

This corresponds to graph B, an open circle at -1/2 and an arrow pointing to the right.

6. $\frac{12}{5} - \frac{x}{5} \le x$

Add $\frac{x}{5}$ to both sides:

$\frac{12}{5} \le x + \frac{x}{5}$

$\frac{12}{5} \le \frac{6x}{5}$

Multiply both sides by 5:

$12 \le 6x$

Divide by 6:

$2 \le x$ or $x \ge 2$

This corresponds to graph C. However, graph C was already matched to inequality

3. There is an error. Re-examine inequality 3:

$5x+4 \ge 7x \implies 4 \ge 2x \implies 2 \ge x$.

Thus, $x \le 2$. This matches graph C, a closed circle at 2, and the line extending to the left.

Re-examine inequality 6: $\frac{12}{5} - \frac{x}{5} \le x$ becomes $12 - x \le 5x \implies 12 \le 6x \implies 2 \le x$ or $x \ge 2$. Thus, x is greater than or equal to

2. This must be graph E, which has a solid circle at 2 and a line going to the right.

3. Final Answer

1. F

2. A

3. C

4. D

5. B

6. E