To solve the inequality \( y \leq -\frac{2}{3}x + 1 \) by graphing, we first need to understand the corresponding line equation \( y = -\frac{2}{3}x + 1 \).
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Identify the intercepts:
- Y-intercept: Set \( x = 0 \): \[ y = -\frac{2}{3}(0) + 1 = 1 \] So, the y-intercept is \( (0, 1) \).
- X-intercept: Set \( y = 0 \): \[ 0 = -\frac{2}{3}x + 1 \] \[ \frac{2}{3}x = 1 \] \[ x = \frac{3}{2} = 1.5 \] So, the x-intercept is \( (1.5, 0) \).
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Plot the line:
- The line has a negative slope of \( -\frac{2}{3} \), meaning it will slope downwards as you move from left to right. The line is dashed since the inequality does not include \( = \).
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Shade the appropriate region:
- Since the inequality is \( y \leq ... \), we shade the area below the line.
Looking at the choices given:
- Response (a) correctly describes a dashed line slanting downwards from left to right, with the line intercepting the y-axis at \( (0, 1) \) and the x-axis at \( (1.5, 0) \). The area below the line is shaded, which corresponds to the inequality \( y \leq -\frac{2}{3}x + 1 \).
Therefore, the correct response is:
a. 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: \( (0, 1) \) and \( (1.5, 0) \).