To graph the system of inequalities for the problem, follow these steps:

### 1. Identify the Inequality to Graph

The main inequality from the problem is:

\[
8x + 5y \leq 800
\]

This represents the budget constraint, where:
- $x$ is the number of pounds of meat.
- $y$ is the number of pounds of tofu.

### 2. Determine the Intercepts

Find the intercepts by setting $x = 0$ and $y = 0$ separately:

- **Y-intercept**: Set $x = 0$:
  \[
  8(0) + 5y = 800 \quad \Rightarrow \quad y = \frac{800}{5} = 160
  \]
  So the y-intercept is $(0, 160)$.

- **X-intercept**: Set $y = 0$:
  \[
  8x + 5(0) = 800 \quad \Rightarrow \quad x = \frac{800}{8} = 100
  \]
  So the x-intercept is $(100, 0)$.

### 3. Plot the Intercepts

On a coordinate plane:
- Plot the point $(0, 160)$ on the y-axis.
- Plot the point $(100, 0)$ on the x-axis.

### 4. Draw the Boundary Line

Draw a straight line through the points $(0, 160)$ and $(100, 0)$. This line represents the equation $8x + 5y = 800$, which is the boundary of the inequality.

### 5. Shade the Feasible Region

Since the inequality is $8x + 5y \leq 800$, shade the region **below and to the left** of the line. This shaded region represents all the possible combinations of $x$ (pounds of meat) and $y$ (pounds of tofu) that satisfy the inequality.

### 6. Consider Non-negativity Constraints

Remember that $x \geq 0$ and $y \geq 0$, meaning you only consider the first quadrant where both $x$ and $y$ are positive. This limits the feasible region to the area below the line within the first quadrant.

### 7. Verify the Region

To ensure you've shaded the correct region, you can pick a test point within the shaded area (e.g., $(50, 50)$) and substitute it into the inequality:

\[
8(50) + 5(50) = 400 + 250 = 650 \quad \text{which is less than } 800
\]

Since the inequality holds true, the region you've shaded is correct.

### Graph

The graph will show:
- A line passing through $(0, 160)$ and $(100, 0)$.
- The area below this line in the first quadrant shaded, representing the feasible region.

Would you like a detailed visual of this graph?

---

### Related Questions

1. How would the graph change if the inequality was strict ($<$ instead of $\leq$)?
2. What happens if the cost of tofu changes to $6 per pound?
3. How do you graph a system of inequalities with more than two variables?
4. Can you use a calculator to graph inequalities? If so, how?
5. How do you interpret the shaded region in real-life terms?
6. How would adding another constraint change the graph?
7. How can you determine the vertices of the feasible region?
8. What is the significance of the boundary line in linear programming?

### Tip

When graphing inequalities, always make sure to clearly distinguish between solid lines (for $\leq$ or $\geq$) and dashed lines (for $<$ or $>$).

Learn how to graph the linear inequality 8x + 5y <= 800 with a detailed step-by-step guide. This problem involves graphing linear inequalities, finding intercepts, and shading the feasible region. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation