Question

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, 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
Attempt 4 out of 5

Point: left parenthesis

Answer 1

To solve the system of inequalities graphically, we need to graph each inequality on the coordinate plane. Let's break it down step by step:

1. **Graph the first inequality:**
\[
y < -\frac{1}{2}x + 4
\]
- This is a linear equation of the form \(y = mx + b\), where the slope (\(m\)) is \(-\frac{1}{2}\) and the y-intercept (\(b\)) is \(4\).
- Plot the line \(y = -\frac{1}{2}x + 4\). Since the inequality is strict (<), the line will be dashed (indicating that points on the line are not included in the solution set).
- Shade the area below the line since we are looking for \(y\) values that are less than the line.

2. **Graph the second inequality:**
\[
y \leq x + 7
\]
- This linear equation has a slope of \(1\) and a y-intercept of \(7\).
- Plot the line \(y = x + 7\). Since the inequality is less than or equal to (≤), the line will be solid (indicating that points on the line are included in the solution set).
- Shade the area below the line because we are interested in \(y\) values that are less than or equal to the line.

3. **Determine the solution set:**
- The solution set consists of the region where the shaded areas from both inequalities overlap.

4. **Finding a point in the solution set:**
One easy point to check is (0, 0):
- For the first inequality:
\[
0 < -\frac{1}{2}(0) + 4 \implies 0 < 4 \quad \text{(True)}
\]
- For the second inequality:
\[
0 \leq 0 + 7 \implies 0 \leq 7 \quad \text{(True)}
\]
Thus, the point (0, 0) satisfies both inequalities, making it part of the solution set.

So, the coordinates of a point in the solution set are:
\[
\text{Point: } (0, 0)
\]