Question

Solve the inequality y ≤− 2/3 x + 1
by graphing.(1 point)
Responses

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: 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 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.

b.
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.
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.

c.
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 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.

d.
A coordinate plane with 4 quadrants shows a graphed dashed line slanting upward from left to right. The line intersects the x-axis at negative 1.5, and it intersects the y-axis at 1. The area below and to the right 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 negative 1.5 comma 0 right parenthesis.
Image with alt text: A coordinate plane with 4 quadrants shows a graphed dashed line slanting upward from left to right. The line intersects the x-axis at negative 1.5, and it intersects the y-axis at 1. The area below and to the right 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 negative 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 understand the corresponding line equation \( y = -\frac{2}{3}x + 1 \).

1. **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 \( (1.5, 0) \).

2. **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 \( = \).

3. **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 \( (1.5, 0) \).