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.