Sketch the region that corresponds to the given inequality. (Select Update Graph to see your response plotted on the screen. Select the Submit button to grade your response.)

2*x* − 4*y* ≤ 5

InequalityTwo Points 

(*x*, *y*)

 on the Graphed LinePoint 

(*x*, *y*)

 in Shaded Region

2*x* − 4*y* ≤ 5

   , 0 




 0,   
  
 
  
Say whether the region is bounded or unbounded.

The region is bounded.The region is unbounded.    

Find the coordinates of all corner points (if any). (If an answer does not exist, enter DNE.)

(*x*, *y*) =  
 
 

We are tasked with sketching the region corresponding to the inequality:

\[
2x - 4y \leq 5
\]

### Step 1: Write the inequality in slope-intercept form.

First, let's convert the given inequality into slope-intercept form (\(y = mx + b\)) for easier graphing:

\[
2x - 4y \leq 5
\]

Subtract \(2x\) from both sides:

\[
-4y \leq -2x + 5
\]

Now, divide the entire inequality by \(-4\). Remember, dividing by a negative number flips the inequality sign:

\[
y \geq \frac{1}{2}x - \frac{5}{4}
\]

This is the equation of a line with slope \(\frac{1}{2}\) and \(y\)-intercept \(-\frac{5}{4}\).

### Step 2: Graph the boundary line.

The boundary line is:

\[
y = \frac{1}{2}x - \frac{5}{4}
\]

This line has a slope of \(\frac{1}{2}\), so it rises by 1 unit for every 2 units it moves to the right. The \(y\)-intercept is \(-\frac{5}{4}\), so the line crosses the \(y\)-axis at this point.

### Step 3: Determine the shaded region.

Since the inequality is \(y \geq \frac{1}{2}x - \frac{5}{4}\), we will shade the region **above** the line. The shaded region represents all the points where \(y\) is greater than or equal to \(\frac{1}{2}x - \frac{5}{4}\).

### Step 4: Test a point to confirm the shading.

A common test point is \((0, 0)\) because it's easy to calculate. Substituting \(x = 0\) and \(y = 0\) into the inequality:

\[
2(0) - 4(0) \leq 5 \implies 0 \leq 5
\]

Since this statement is true, the point \((0, 0)\) lies within the shaded region.

### Step 5: Determine whether the region is bounded or unbounded.

To decide whether the region is bounded or unbounded, observe that this is a linear inequality with no upper or lower constraints. Since the shaded region extends infinitely in some directions, **the region is unbounded**.

### Step 6: Find the corner points (if any).

This inequality represents a half-plane bounded by a straight line. Since there are no additional constraints, the region extends infinitely and has no finite corner points.

So the answer for corner points is:

\[
(x, y) = \text{DNE}
\]

### Summary of answers:

- **Shaded region**: The area above the line \(y = \frac{1}{2}x - \frac{5}{4}\).
- **Bounded or unbounded**: The region is **unbounded**.
- **Corner points**: DNE (no corner points).

Let me know if you'd like me to clarify or provide a graph! Also, here are some additional questions to deepen your understanding:

1. How do we interpret the slope of the boundary line geometrically?
2. What changes in the graph if we switch the inequality to \(2x - 4y \geq 5\)?
3. How would we determine if a region is bounded if given a system of inequalities?
4. Can we have a bounded region without any corner points?
5. How would introducing another inequality affect the shape of the region?

**Tip:** When graphing inequalities, always test a point outside of the line to ensure you shade the correct region.

Sketch the region that corresponds to the given inequality. (Select Update Graph to see your response plotted on the screen. Select the Submit button to grade your response.)

2*x* − 4*y* ≤ 5

Inequality

(*x*, *y*) on the Graphed Line
Point (*x*, *y*) in Shaded Region

Say whether the region is bounded or unbounded.

The region is bounded. The region is unbounded.

Find the coordinates of all corner points (if any). (If an answer does not exist, enter DNE.)

(*x*, *y*) =

Learn how to graph and analyze the linear inequality 2x - 4y ≤ 5. Understand how to shade the correct region, determine if the region is bounded or unbounded, and find corner points. This problem covers key concepts in graphing inequalities and coordinate geometry, suitable for Grades 9-11 students.

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