Question

which of the points on the graph is a solution for the stimulus inequalities why is less than 4x - 2 and Y is less than equal to -1/3 x + 2 Enter the x value followed by the Y value

Answer 1

To find a solution that satisfies the inequalities \( y < 4x - 2 \) and \( y \leq -\frac{1}{3}x + 2 \), we can look for a point where both conditions are true.

1. **Graph the inequalities**:
- For \( y < 4x - 2 \):
- This is a line with a slope of 4 and a y-intercept of -2. The area below this line represents the solutions for this inequality (since it is a "less than" inequality).

- For \( y \leq -\frac{1}{3}x + 2 \):
- This is a line with a slope of -\(\frac{1}{3}\) and a y-intercept of 2. The area below this line, including the line itself, represents the solutions for this inequality (since it is a "less than or equal to" inequality).

2. **Find the intersection point**:
To find where both inequalities are satisfied, we can first find the intersection of the two lines:
\[
4x - 2 = -\frac{1}{3}x + 2
\]
Multiply the entire equation by 3 to eliminate the fraction:
\[
12x - 6 = -x + 6
\]
Rearranging gives:
\[
12x + x = 6 + 6 \\
13x = 12 \\
x = \frac{12}{13}
\]

Now, substituting \(x = \frac{12}{13}\) back into either equation to find \(y\):
Using \(y = 4x - 2\):
\[
y = 4\left(\frac{12}{13}\right) - 2 \\
y = \frac{48}{13} - \frac{26}{13} \\
y = \frac{22}{13}
\]

Thus, the intersection point is \(\left(\frac{12}{13}, \frac{22}{13}\right)\).

3. **Determine if it's a solution**:
Now, choose points around this intersection to check which satisfies both inequalities. Let's take the point \(\left(1, 0\right)\) to see if it fits:

- For \(y < 4x - 2\):
- \(0 < 4(1) - 2\)
- \(0 < 2\) (True)

- For \(y \leq -\frac{1}{3}x + 2\):
- \(0 \leq -\frac{1}{3}(1) + 2\)
- \(0 \leq -\frac{1}{3} + 2\)
- \(0 \leq \frac{5}{3}\) (True)

Thus, the point (1, 0) is a solution to the system of inequalities.

So, one of the solutions would be:
\[
\text{(1, 0)}
\]