To solve the inequalities provided and to represent their solutions graphically, let's start by identifying and solving each inequality. Then, we'll describe how to graph the solutions and shade the appropriate regions.

1. $4x \leq 4$
2. $2y + 3 \leq x + 2$
3. $3 \leq x - y - 5$
4. $2x - 7y \geq 3$
5. $3x + y \geq 5$
6. $6 \leq 7x + 2y$
7. $y \geq x - 5$

Let's solve each inequality step-by-step:

1. $4x \leq 4$

Divide both sides by 4: $x \leq 1$

2. $2y + 3 \leq x + 2$

Subtract 2 from both sides: $2y + 1 \leq x$ Rearrange to isolate $y$: $y \leq \frac{x - 1}{2}$

3. $3 \leq x - y - 5$

Add 5 to both sides: $8 \leq x - y$ Rearrange to isolate $y$: $y \leq x - 8$

4. $2x - 7y \geq 3$

Rearrange to isolate $y$: $-7y \geq 3 - 2x$ Divide by -7 (reversing the inequality sign): $y \leq \frac{2x - 3}{7}$

5. $3x + y \geq 5$

Rearrange to isolate $y$: $y \geq 5 - 3x$

6. $6 \leq 7x + 2y$

Rearrange to isolate $y$: $2y \geq 6 - 7x$ Divide by 2: $y \geq 3.5 - \frac{7x}{2}$

7. $y \geq x - 5$

This inequality is already solved for $y$.

Graphical Representation

To graph these inequalities, follow these steps for each inequality:

1. Graph the boundary line (treat the inequality as an equality).
2. Determine the shading:
   • For $y \leq$ inequalities, shade below the line.
   • For $y \geq$ inequalities, shade above the line.
   • For $x \leq$ or $x \geq$ inequalities, shade to the left or right of the line respectively.

Let's describe the graphing for each:

1. $x \leq 1$: A vertical line at $x = 1$. Shade to the left.
2. $y \leq \frac{x - 1}{2}$: A line with a slope of 1/2 and y-intercept of -1/2. Shade below.
3. $y \leq x - 8$: A line with a slope of 1 and y-intercept of -8. Shade below.
4. $y \leq \frac{2x - 3}{7}$: A line with a slope of 2/7 and y-intercept of -3/7. Shade below.
5. $y \geq 5 - 3x$: A line with a slope of -3 and y-intercept of 5. Shade above.
6. $y \geq 3.5 - \frac{7x}{2}$: A line with a slope of -7/2 and y-intercept of 3.5. Shade above.
7. $y \geq x - 5$: A line with a slope of 1 and y-intercept of -5. Shade above.

By plotting these lines and shading the corresponding regions, you can identify the solution set for each inequality. If you have graphing paper or a graphing tool, you can plot these lines and shade the regions accordingly.

Do you want more detailed steps on graphing or have any questions about solving these inequalities?

Here are some related questions for further exploration:

1. How do you solve a system of linear inequalities?
2. What is the feasible region in linear programming?
3. How do you determine if a point is in the solution set of an inequality?
4. What are the differences between strict and non-strict inequalities in graphing?
5. How do you interpret the solution to a system of inequalities in real-world applications?
6. What are some common methods to solve systems of inequalities algebraically?
7. How does graphing a system of inequalities differ from graphing a single inequality?
8. Can systems of inequalities have no solution? How do you identify this graphically?

Tip: When graphing inequalities, always double-check the direction of shading to ensure accurate representation of the solution set.