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Given the graph of the inequality in two variables, the numbers are (0,-1) and (2,0) fill in the missing pieces of the solution set. (2 points) _x+ _y<_

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To find the equation represented by the inequality that includes the points (0, -1) and (2, 0), we first determine the equation of the line that passes through these two points.

  1. Find the slope (m) of the line: \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - (-1)}{2 - 0} = \frac{1}{2} \]

  2. Use the point-slope form of the line: Using point (0, -1): \[ y - y_1 = m(x - x_1) \Rightarrow y - (-1) = \frac{1}{2}(x - 0) \] Simplifying: \[ y + 1 = \frac{1}{2}x \Rightarrow y = \frac{1}{2}x - 1 \]

  3. Convert to standard form: Rearranging gives us: \[ -\frac{1}{2}x + y + 1 = 0 \Rightarrow x - 2y - 2 = 0 \Rightarrow x - 2y = 2 \]

  4. Create the inequality: To determine the correct inequality, test one of the points against this line. We can choose (0, 0): \[ 0 - 2(0) < 2 \Rightarrow 0 < 2 \quad \text{(true)} \] Therefore, the solution set is above the line.

Thus, the inequality is: \[ x - 2y < 2 \] So the final filled-in form is: \[ 1x + (-2)y < 2 \] In summary, the solution set for the inequality is: \[ x - 2y < 2 \]

wrong, theres supposed to be a number next to the x

I apologize for the oversight in my previous message. Let's reanalyze the information and correctly express the inequality in the form you're asking for.

Based on the points \( (0, -1) \) and \( (2, 0) \), we determined a line with the slope of \( \frac{1}{2} \).

To write this in the form you specified:

  1. Equation of the line:
    We calculated the line equation: \[ y = \frac{1}{2}x - 1 \]

  2. To put this in standard form \( Ax + By < C \):
    To move \(y\) to one side: \[ -\frac{1}{2} x + y < -1 \]

  3. Multiply through by -2 (and reverse the inequality): \[ x - 2y > 2 \] or in the desired form: \[ 1x + (-2)y < 2 \]

This means the correct inequality in the format you wanted is: \[ 1x + (-2)y < 2 \]

Again, I apologize for the confusion earlier. Thank you for your patience!

thats correct thank you

You're welcome! I'm glad we got it right. If you have any more questions or need further assistance, feel free to ask!