Question

Sketch the solution for the system of inequalities.
y > -x + 2
y >/ 1/3x - 2

Answer 1

To solve the system of inequalities given by:

1. \( y > -x + 2 \)
2. \( y \geq \frac{1}{3}x - 2 \)

we will sketch the solution by following these steps:

### Step 1: Graph each inequality

**1. Graph the line for \( y = -x + 2 \)**:
- This is a straight line with a slope of -1 and a y-intercept of 2.
- To graph it, you can plot two points:
- When \( x = 0 \), \( y = 2 \) (the intercept).
- When \( x = 2 \), \( y = 0 \) (a second point).
- Draw a dashed line for this inequality since the inequality is strict (">").

**2. Graph the line for \( y = \frac{1}{3}x - 2 \)**:
- This line has a slope of \( \frac{1}{3} \) and a y-intercept of -2.
- Plot two points:
- When \( x = 0 \), \( y = -2 \).
- When \( x = 3 \), \( y = -1 \) (another point).
- Draw a solid line for this inequality since the inequality is non-strict ("≥").

### Step 2: Shade the regions

**1. For the inequality \( y > -x + 2 \)**:
- Shade the area above the dashed line (not including the line itself).

**2. For the inequality \( y \geq \frac{1}{3}x - 2 \)**:
- Shade the area above (and including) the solid line.

### Step 3: Identify the solution region

The solution to the system of inequalities is the area where the shaded regions for both inequalities intersect.

### Step 4: Final sketch

- Draw both lines as described above with appropriate shading.
- The region that satisfies both inequalities will be above the dashed line and also above and on the solid line.

This sketch represents the solution set for the system of inequalities.