https://file.aiquickdraw.com/mathbot/file/1729718907381te5d9673.jpg

We are asked to identify the correct graph for the system of inequalities:

\[
y < \frac{1}{2}x - 2 \quad \text{and} \quad y \leq -2x + 4
\]

### Step 1: Analyze the first inequality
\[
y < \frac{1}{2}x - 2
\]
- This is a linear inequality with a slope of \( \frac{1}{2} \) and a y-intercept of \( -2 \).
- The graph of the boundary line \( y = \frac{1}{2}x - 2 \) is dashed, indicating that the inequality is strict (it does not include the line itself).
- The shading will be below this line because of the "less than" sign.

### Step 2: Analyze the second inequality
\[
y \leq -2x + 4
\]
- This is another linear inequality, but this time with a slope of \( -2 \) and a y-intercept of \( 4 \).
- The boundary line \( y = -2x + 4 \) is solid, indicating the inequality includes the line.
- The shading will be below the line because of the "less than or equal to" sign.

### Step 3: Look for the correct graph
We need a graph that shows:
1. A dashed line for \( y = \frac{1}{2}x - 2 \), with shading below it.
2. A solid line for \( y = -2x + 4 \), with shading below it.
3. The solution region is where both shaded areas overlap.

Now, examining the provided images:
- **The first graph** shows both lines, but the shading and type of lines are incorrect.
- **The second graph** displays the correct solid and dashed lines, but the shaded region is incorrect.
- **The third graph** shows the correct lines with shading below both, but the solution region is not correct.
- **The fourth graph** is correct. It has:
  - A dashed line for \( y = \frac{1}{2}x - 2 \),
  - A solid line for \( y = -2x + 4 \),
  - And shading in the correct region (below both lines).

### Conclusion:
The fourth graph represents the solution to the system of inequalities.

Would you like further details on any part of this explanation?

Here are five related questions to deepen your understanding:
1. How do you determine which side of the boundary line to shade in a linear inequality?
2. What is the difference between a dashed line and a solid line in the graph of an inequality?
3. How can you check whether a point is a solution to a system of inequalities?
4. How do the slopes of two lines affect the region where their solutions overlap?
5. Can the solution to a system of inequalities ever be an empty set?

**Tip**: Always check both the line type (dashed or solid) and the shading direction when interpreting inequality graphs!

Which graph shows the solution to this system of inequalities? y < (1/2)x - 2 and y ≤ -2x + 4

Identify the correct graph for the system of inequalities y < (1/2)x - 2 and y ≤ -2x + 4. This problem involves graphing linear inequalities and finding the solution region where both inequalities overlap. Suitable for students in Grades 8-10.

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