To determine which inequality could represent the given table of values, we need to find the equation of the line that passes through the provided points. Then, we can set up an inequality based on that line.
First, we can calculate the slope and y-intercept of the line by using two points from the table. Let's use the points \((-6, -13)\) and \(4, 2\).
- Calculate the slope \(m\):
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - (-13)}{4 - (-6)} = \frac{2 + 13}{4 + 6} = \frac{15}{10} = \frac{3}{2} \]
- Now we can use the slope-intercept form \(y = mx + b\) to find the y-intercept \(b\). We can substitute one of the points into the equation. Let's use the point \((4, 2)\):
\[ 2 = \frac{3}{2}(4) + b \]
Calculating \(\frac{3}{2} \times 4\):
\[ 2 = 6 + b \implies b = 2 - 6 = -4 \]
Thus, the equation of the line is:
\[ y = \frac{3}{2}x - 4 \]
Now, since the original question asks about inequalities, we consider whether we want the area above or below this line. Since all values in the table represent points about the inequality being greater than or equal to (the boundary line), we can represent it as:
\[ y \geq \frac{3}{2}x - 4 \]
Now, let's check which of the given options matches:
A. \(y \geq 32x - 4\)
B. \(y \geq 23x + 83\)
C. \(y \geq 45x - 2\)
D. \(y \geq 54x + 32\)
It seems none of these options are consistent with \(y \geq \frac{3}{2}x - 4\). Therefore, it is likely that there may have been a mistake in the question's setup or the provided options.
Upon further inspection of the slope we calculated, it's apparent that we should verify if any of the proposed inequalities equate to \(y = \frac{3}{2}x - 4\):
\(\frac{3}{2} = 1.5\) — Thus, \(\frac{3}{2} = \frac{15}{10} = \frac{45}{30}\) would equate to none of the choices but the closest option for the y-value intercept yielded should be recalculated based on that.
If I missed any point that could simplify into representing any listed inequalities, let me know for validation!
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