Solve the following system of inequalities graphically on the set of axes below. State the coordinates of a point in the solution set.
y, is less than or equal to, 2, x, minus, 1
y≤2x−1
y, is greater than or equal to, minus, x, plus, 2
y≥−x+2

To solve the system of inequalities graphically, we need to plot the lines corresponding to each inequality and then determine the region where the inequalities overlap. Here are the steps to solve the given system:

### Step 1: Graph the Boundary Lines

1. **Graph the line for \(y \leq 2x - 1\):**
   - This line has a slope of 2 and a y-intercept of -1.
   - To graph it, start at the point (0, -1) on the y-axis.
   - From (0, -1), use the slope of 2 (rise over run) to go up 2 units and right 1 unit. Plot another point at (1, 1).
   - Connect these points with a straight line. This line represents \(y = 2x - 1\).

2. **Graph the line for \(y \geq -x + 2\):**
   - This line has a slope of -1 and a y-intercept of 2.
   - To graph it, start at the point (0, 2) on the y-axis.
   - From (0, 2), use the slope of -1 (rise over run) to go down 1 unit and right 1 unit. Plot another point at (1, 1).
   - Connect these points with a straight line. This line represents \(y = -x + 2\).

### Step 2: Determine the Shaded Regions

- **For the inequality \(y \leq 2x - 1\):**
  - The region below the line \(y = 2x - 1\) should be shaded because the inequality is less than or equal to (≤).

- **For the inequality \(y \geq -x + 2\):**
  - The region above the line \(y = -x + 2\) should be shaded because the inequality is greater than or equal to (≥).

### Step 3: Find the Intersection Point

To find the point where the two lines intersect, we set the equations equal to each other:

\[
2x - 1 = -x + 2
\]

Solve for \(x\):

\[
2x + x = 2 + 1
\]
\[
3x = 3
\]
\[
x = 1
\]

Now substitute \(x = 1\) into either equation to find \(y\):

\[
y = 2(1) - 1 = 1
\]

So, the intersection point is \((1, 1)\).

### Step 4: Identify a Point in the Solution Set

The solution set is the region where both shaded areas overlap. A point in this overlapping region is (1, 1), which is also the intersection point. Another possible point is (0, 2).

### Conclusion

The coordinates of a point in the solution set are \((1, 1)\) or \((0, 2)\).

Would you like a step-by-step visualization of the graph, or do you have any questions?

### Related Questions:

1. How do you graph systems of inequalities with more than two lines?
2. What is the importance of the slope in linear equations?
3. How can we solve systems of linear inequalities algebraically?
4. How does changing the inequality symbol affect the solution region?
5. Can a system of linear inequalities have an infinite number of solutions?

**Tip:** Always label your axes and plot points carefully when graphing inequalities to avoid mistakes in determining the solution region.

Solve the following system of inequalities graphically on the set of axes below. State the coordinates of a point in the solution set.

 y ≤ 2x − 1
 y ≥ −x + 2

Learn how to solve a system of inequalities graphically, including y ≤ 2x − 1 and y ≥ −x + 2. This guide covers graphing techniques, identifying intersection points, and determining the solution region. Suitable for students in Grades 8-10.

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