Solve each system of inequalities by graphing. x ≤ 3, x + y > 2

To solve the system of inequalities by graphing, we need to graph each inequality separately and then find the region where the solutions to both inequalities overlap.

### Step 1: Graph the inequality \( x \leq 3 \)

- The inequality \( x \leq 3 \) represents all points where \( x \) is less than or equal to 3. 
- The boundary line is \( x = 3 \), which is a vertical line passing through \( x = 3 \).
- Since the inequality is \( \leq \), we shade the region to the left of the line \( x = 3 \), including the line itself.

### Step 2: Graph the inequality \( x + y > 2 \)

- The inequality \( x + y > 2 \) can be rewritten in slope-intercept form as \( y > -x + 2 \).
- The boundary line is \( y = -x + 2 \), which has a slope of -1 and a y-intercept of 2.
- Since the inequality is \( > \), we draw the line as a dashed line (indicating that points on the line are not included in the solution), and shade the region above the line where \( y \) is greater than \( -x + 2 \).

### Step 3: Identify the solution region

- The solution to the system of inequalities is the region where the shaded areas from both inequalities overlap.

### Graphical Representation:

The graph would look like this:

1. A vertical line at \( x = 3 \), with the region to the left shaded.
2. A dashed line representing \( y = -x + 2 \), with the region above the line shaded.
3. The solution region is where these two shaded areas intersect.

### Conclusion:

The solution region will be the area to the left of the line \( x = 3 \) and above the line \( y = -x + 2 \).

Would you like further details, a graph to visualize, or any other questions? Here are some related questions you might consider:

1. How do you determine the boundary line for a given inequality?
2. What does it mean when an inequality includes an "equal to" part, such as \( \leq \) or \( \geq \)?
3. How does the slope of a line affect the region that satisfies an inequality?
4. What would happen if the inequality were \( x + y \geq 2 \) instead of \( x + y > 2 \)?
5. How can you verify the solution to a system of inequalities algebraically?

**Tip:** When graphing inequalities, always remember to check whether the line should be solid or dashed depending on whether the inequality includes equality (≥, ≤) or not (>, <).

Learn how to solve the system of inequalities x ≤ 3 and x + y > 2 by graphing. This problem involves graphing each inequality separately, understanding boundary lines, and shading regions to find overlapping solutions. Suitable for Grades 7-9.

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