Question

Graph the inequality on the axes below.
y, is greater than, minus, start fraction, 2, divided by, 3, end fraction, x, plus, 5
y>−
3
2
​
x+5
Can you pls tell me the 2 points also they have to intercept the line so were the points on the lines are i need points outside it that go through them and pls tell me witch side is getting shaded

Answer 1

To graph the inequality \( y > -\frac{2}{3}x + 5 \), you'll first want to graph the line represented by the equation \( y = -\frac{2}{3}x + 5 \). Here are the steps to find points on the line, as well as points outside the line, and what side to shade.

### Step 1: Finding the Intercepts

1. **Y-Intercept**: Set \( x = 0 \).
\[
y = -\frac{2}{3}(0) + 5 = 5
\]
So the y-intercept is \( (0, 5) \).

2. **X-Intercept**: Set \( y = 0 \).
\[
0 = -\frac{2}{3}x + 5
\]
Rearranging gives:
\[
\frac{2}{3}x = 5 \implies x = 5 \cdot \frac{3}{2} = \frac{15}{2} = 7.5
\]
So the x-intercept is \( \left(7.5, 0\right) \).

### Summary of Points on the Line

- **Y-Intercept**: \( (0, 5) \)
- **X-Intercept**: \( (7.5, 0) \)

### Step 2: Finding Points Outside the Line

To find points that satisfy the inequality \( y > -\frac{2}{3}x + 5 \), select points above the line. Here are two points outside the line:

1. **Point One**: \( (0, 6) \) (above the y-intercept)
- \( 6 > -\frac{2}{3}(0) + 5 \) → \( 6 > 5 \) (True)

2. **Point Two**: \( (2, 6) \)
- \( 6 > -\frac{2}{3}(2) + 5 \) → \( 6 > -\frac{4}{3} + 5 \) → \( 6 > \frac{15}{3} - \frac{4}{3} \) → \( 6 > \frac{11}{3} \) (True)
- Note: \( \frac{11}{3} \approx 3.67 \), so this point is also above the line.

### Step 3: Determine the Shading

Since the inequality is \( y > -\frac{2}{3}x + 5 \), you will shade above the line.

### Conclusion

- **Points on the line**: \( (0, 5) \) and \( (7.5, 0) \)
- **Points above the line**: \( (0, 6) \) and \( (2, 6) \)
- **Shaded Region**: Above the line \( y = -\frac{2}{3}x + 5 \)

Using this information, you can plot the graph accurately on a coordinate system.