To find a point that satisfies the inequalities \( y > 4x - 2 \) and \( y \geq -13x + 2 \), we can analyze these inequalities graphically or mathematically.
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Rewrite the inequalities:
- For the first inequality \( y > 4x - 2 \): This represents the area above the line \( y = 4x - 2 \).
- For the second inequality \( y \geq -13x + 2 \): This represents the area on or above the line \( y = -13x + 2 \).
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Find the intersection point of the lines:
Set \( 4x - 2 = -13x + 2 \):
\[ 4x + 13x = 2 + 2 \] \[ 17x = 4 \] \[ x = \frac{4}{17} \]
Now substitute \( x = \frac{4}{17} \) back to find \( y \):
\[ y = 4\left(\frac{4}{17}\right) - 2 = \frac{16}{17} - \frac{34}{17} = -\frac{18}{17} \]
Check with the second line:
\[ y = -13\left(\frac{4}{17}\right) + 2 = -\frac{52}{17} + \frac{34}{17} = -\frac{18}{17} \]
Thus the intersection point is \( \left(\frac{4}{17}, -\frac{18}{17}\right) \).
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Determine a solution point:
Since we need one of the solutions of the inequalities, we can try values slightly above and to the right of this intersection point to ensure they satisfy both inequalities. For example, let's try \( x = 1 \):
- For \( x = 1 \):
\[ y > 4(1) - 2 = 2 \quad \text{(consider } y > 2\text{)} \]
\[ y \geq -13(1) + 2 = -11 \quad \text{(consider } y \geq -11\text{)} \]
Choose \( y = 3 \):
This choice gives us:
- \( y > 2 \): True (since \( 3 > 2 \))
- \( y \geq -11 \): True (since \( 3 \geq -11 \))
Therefore, the point \( (1, 3) \) is a solution.
Thus, the answer format your required \( x \)-value followed by the \( y \)-value becomes:
\[ \boxed{1 , 3} \]
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