Question

Solve the inequality y≤−2/3x+1 by graphing.(1 point) Responses Image with alt text: A coordinate plane with 4 quadrants shows a graphed dashed line gradually slanting downward from left to right. The line intersects the x-axis at 1.5, and it intersects the y-axis at 1. The area below and to the left of the line is shaded. The coordinates where the line intersects the axes are as follows: left parenthesis 0 comma 1 right parenthesis and left parenthesis 1.5 comma 0 right parenthesis. Image with alt text: A coordinate plane with 4 quadrants shows a graphed line sloping gradually downward from left to right. The line crosses the x-axis at negative 1.5 and the y-axis at 1. The area below and to the right of the line is shaded. The line intersects the x-axis at coordinates left parenthesis 0 comma 1 right parenthesis and left parenthesis negative 1.5 comma 0 right parenthesis. Image with alt text: A coordinate plane with 4 quadrants shows a graphed line sloping gradually downward from left to right. The line crosses the x-axis at 1.5 and the y-axis at 1. The area below and to the left of the line is shaded. The line intersects the x-axis at coordinates left parenthesis 0 comma 1 right parenthesis and the y-axis at left parenthesis 1.5 comma 0 right parenthesis.

Answer 1

To solve the inequality \( y \leq -\frac{2}{3}x + 1 \) by graphing, we first need to graph the related equation \( y = -\frac{2}{3}x + 1 \).

Steps to Graph the Equation:

1. Identify the intercepts:

   • Y-intercept: Set \( x = 0 \). \[ y = -\frac{2}{3}(0) + 1 = 1 \quad \Rightarrow \quad (0, 1) \]
   • X-intercept: Set \( y = 0 \). \[ 0 = -\frac{2}{3}x + 1 \quad \Rightarrow \quad \frac{2}{3}x = 1 \quad \Rightarrow \quad x = \frac{3}{2} = 1.5 \quad \Rightarrow \quad (1.5, 0) \]

2. Plot the points:

   • Plot the points \( (0, 1) \) and \( (1.5, 0) \) on the coordinate plane.

3. Draw the line:

   • Since the inequality is \( \leq \), we will draw a solid line through these points because it includes the points on the line (the inequality includes equality).

4. Shade the appropriate region:

   • Since the inequality is \( y \leq -\frac{2}{3}x + 1 \), we shade the region below the line.

Conclusion:

From your description, the graph that meets the conditions of the inequality, showing a solid line (not dashed), with the correct shading below the line and intersecting the axes at \( (0, 1) \) and \( (1.5, 0) \), is what represents the inequality.

The correct response would look similar to the third description (but with a solid line) since it shows the line sloping downward from left to right, and shading that reflects the region where \( y \) is less than or equal to the line equation.