Question

Which of the points on the graph is a solution for the simultaneous inequalities y>4x−2 and y≥−1/3x+2 ? Enter the x-value followed by the y-value. (1 point)
The points are: (-1, 5)
(0.923, 1.6)
(5, 2)
(-4, -2)

Answer 1

To determine which points are solutions to the simultaneous inequalities \( y > 4x - 2 \) and \( y \geq -\frac{1}{3}x + 2 \), we can evaluate each point one at a time.

1. **Point (-1, 5)**:
- For \( y > 4x - 2 \):
\[
5 > 4(-1) - 2 \Rightarrow 5 > -4 - 2 \Rightarrow 5 > -6 \quad \text{(True)}
\]
- For \( y \geq -\frac{1}{3}x + 2 \):
\[
5 \geq -\frac{1}{3}(-1) + 2 \Rightarrow 5 \geq \frac{1}{3} + 2 \Rightarrow 5 \geq \frac{7}{3} \quad \text{(True)}
\]
- Result: **True** for both inequalities.

2. **Point (0.923, 1.6)**:
- For \( y > 4x - 2 \):
\[
1.6 > 4(0.923) - 2 \Rightarrow 1.6 > 3.692 - 2 \Rightarrow 1.6 > 1.692 \quad \text{(False)}
\]
- Result: **False** for the first inequality.

3. **Point (5, 2)**:
- For \( y > 4x - 2 \):
\[
2 > 4(5) - 2 \Rightarrow 2 > 20 - 2 \Rightarrow 2 > 18 \quad \text{(False)}
\]
- Result: **False** for the first inequality.

4. **Point (-4, -2)**:
- For \( y > 4x - 2 \):
\[
-2 > 4(-4) - 2 \Rightarrow -2 > -16 - 2 \Rightarrow -2 > -18 \quad \text{(True)}
\]
- For \( y \geq -\frac{1}{3}x + 2 \):
\[
-2 \geq -\frac{1}{3}(-4) + 2 \Rightarrow -2 \geq \frac{4}{3} + 2 \Rightarrow -2 \geq \frac{4}{3} + \frac{6}{3} \Rightarrow -2 \geq \frac{10}{3} \quad \text{(False)}
\]
- Result: **False** for the second inequality.

The only point that satisfies both inequalities is **(-1, 5)**.

Thus, the solution is:
\[
\boxed{-1 \, 5}
\]