Question

Solve the following system of inequalities graphically on the set of axes below. State the coordinates of a point in the solution set. And put both two points for both inequalities.
y, is less than, minus, one half, x, plus, 4
y<−
1/2
​
x+4
y, is less than or equal to, x, plus, 7
y≤x+7
x
y
x
y
Answer


Point:

Answer 1

dashed or soild line

Answer 2

To solve the system of inequalities graphically, we start by plotting the boundary lines for each inequality.

1. **First Inequality**:
\[ y < -\frac{1}{2}x + 4 \]
The boundary line is \( y = -\frac{1}{2}x + 4 \).
- To find points on this line, we can choose two values for \( x \):

- When \( x = 0 \):
\[ y = -\frac{1}{2}(0) + 4 = 4 \]
(Point: (0, 4))

- When \( x = 4 \):
\[ y = -\frac{1}{2}(4) + 4 = 2 \]
(Point: (4, 2))

This line has a negative slope, so it goes downwards from left to right. The region below the line represents the solution set for this inequality.

2. **Second Inequality**:
\[ y \leq x + 7 \]
The boundary line is \( y = x + 7 \).
- To find points on this line:

- When \( x = 0 \):
\[ y = 0 + 7 = 7 \]
(Point: (0, 7))

- When \( x = -7 \):
\[ y = -7 + 7 = 0 \]
(Point: (-7, 0))

This line has a positive slope, so it goes upwards from left to right. The region below or on this line (since it is less than or equal to) is the solution set for this inequality.

3. **Finding the Overlapping Region**:
The solution to the system of inequalities is the region that is below both lines. You can shade the area under both lines on your graph.

### Points
Here are two points from the solution set:
- **Point 1**: (0, 3) which is below both lines.
- **Point 2**: (-6, 1) which lies in the region defined by both inequalities.

Thus, a point in the solution set could be (0, 3).

### Answer Summary:
Two points:
- From the first inequality: (0, 4) and (4, 2)
- From the second inequality: (0, 7) and (-7, 0)

**Point in the solution set**: (0, 3)