Questions: Which linear inequality is represented by the graph? y<3x+2 y>3x+2 y<1/3x+2 y>1/3x+2

Transcript text: Which linear inequality is represented by the graph? $y<3 x+2$ $y>3 x+2$ $y<\frac{1}{3} x+2$ $y>\frac{1}{3} x+2$

Solution

Solution Steps

Step 1: Identify the slope and y-intercept of the boundary line

The boundary line on the graph passes through the points (0, 2) and (-3, -7). We can use these points to find the slope (m) and y-intercept (b) of the line.

Step 2: Calculate the slope (m)

The slope (m) is calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Using the points (0, 2) and (-3, -7): \[ m = \frac{-7 - 2}{-3 - 0} = \frac{-9}{-3} = 3 \]

Step 3: Determine the y-intercept (b)

The y-intercept (b) is the y-coordinate of the point where the line crosses the y-axis. From the graph, the line crosses the y-axis at (0, 2), so: \[ b = 2 \]

Step 4: Form the equation of the boundary line

Using the slope-intercept form $ y = mx + b $, we get: \[ y = 3x + 2 \]

Step 5: Determine the inequality

The shaded region is above the line, indicating a "greater than" inequality. Since the line is solid, it includes the boundary line, so the inequality is: \[ y \geq 3x + 2 \]

Final Answer

\[ y \geq 3x + 2 \]

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