Question

Graph the system to determine which of the following points represents a solution to the system.

y≥−13x−4
𝑦
≥
−
1
3
𝑥
−
4

y<2x−5
𝑦
<
2
𝑥
−
5

x<4
𝑥
<
4

(1 point)
Responses

(3,−3)
(
3
,
−
3
)
left parenthesis 3 comma negative 3 right parenthesis

(4,−6)
(
4
,
−
6
)
left parenthesis 4 comma negative 6 right parenthesis

(3,1)
(
3
,
1
)
left parenthesis 3 comma 1 right parenthesis

(4,3)

Answer 1

To determine which points satisfy the given system of inequalities, we first need to graph the inequalities and then check each point against the conditions.

1. **Inequalities**:
- \( y \geq -13x - 4 \) (This line is solid because of "≥")
- \( y \geq -\frac{1}{3}x - 4 \) (This line is also solid)
- \( y < 2x - 5 \) (This line is dashed due to "<")
- \( x < 4 \) (This is a vertical dashed line at \( x=4 \))

2. **Graphing the Lines**:
- First, we graph the lines from the first two inequalities (solid lines).
- For \( y = -13x - 4 \), the y-intercept is at -4 and the slope is -13 (very steep).
- For \( y = -\frac{1}{3}x - 4 \), the y-intercept is at -4 and the slope is -1/3 (gentler slope).
- Then we graph the line \( y = 2x - 5 \) (dashed line) with a y-intercept at -5 and a slope of 2.
- Finally, we draw a vertical dashed line at \( x = 4 \).

3. **Finding the Solution Region**:
- The solution region for \( y \geq -13x - 4 \) and \( y \geq -\frac{1}{3}x - 4 \) is above these lines.
- The region for \( y < 2x - 5 \) is below that line.
- The region for \( x < 4 \) is to the left of the dashed line at \( x = 4 \).

4. **Checking Points**:
- **(3, -3)**:
- For \( y \geq -13(3) - 4 \) : Is \(-3 \geq -39 - 4\) (True)
- For \( y \geq -\frac{1}{3}(3) - 4 \) : Is \(-3 \geq -1 - 4\) (True)
- For \( y < 2(3) - 5 \) : Is \(-3 < 6 - 5\) (True)
- For \( x < 4 \) : Is \(3 < 4\) (True)
- **Conclusion**: (3, -3) is a solution.

- **(4, -6)**:
- For \( y \geq -13(4) - 4 \) : Is \(-6 \geq -52 - 4\) (True)
- For \( y \geq -\frac{1}{3}(4) - 4 \) : Is \(-6 \geq -\frac{4}{3} - 4\) (True)
- For \( y < 2(4) - 5 \) : Is \(-6 < 8 - 5\) (True)
- For \( x < 4 \) : Is \(4 < 4\) (False)
- **Conclusion**: (4, -6) is NOT a solution.

- **(3, 1)**:
- For \( y \geq -13(3) - 4 \) : Is \(1 \geq -39 - 4\) (True)
- For \( y \geq -\frac{1}{3}(3) - 4 \) : Is \(1 \geq -1 - 4\) (True)
- For \( y < 2(3) - 5 \) : Is \(1 < 6 - 5\) (False)
- **Conclusion**: (3, 1) is NOT a solution.

- **(4, 3)**:
- For \( y \geq -13(4) - 4 \) : Is \(3 \geq -52 - 4\) (True)
- For \( y \geq -\frac{1}{3}(4) - 4 \) : Is \(3 \geq -\frac{4}{3} - 4\) (True)
- For \( y < 2(4) - 5 \) : Is \(3 < 8 - 5\) (True)
- For \( x < 4 \) : Is \(4 < 4\) (False)
- **Conclusion**: (4, 3) is NOT a solution.

Based on the evaluations, the point that satisfies all inequalities in the system is **(3, -3)**.