Question

on a piece of paper, graph $y < -\frac{2}{5}x + 1$. then determine which answer choice matches the graph you drew. a b c d text description for graph a. graph a b. graph b c. graph c d. graph d

Explanation:

Step1: Analyze the inequality type

The inequality is \( y < -\frac{2}{5}x + 1 \). Since the inequality is strict (\(<\)), the boundary line \( y = -\frac{2}{5}x + 1 \) should be dashed (not solid). Let's check the y - intercept: when \( x = 0 \), \( y=1 \), so the line passes through \((0,1)\). The slope is \( -\frac{2}{5} \), which is negative, so the line should be decreasing.

Step2: Determine the shading region

To find the region where \( y < -\frac{2}{5}x + 1 \), we can test a point not on the line. Let's test the origin \((0,0)\): substitute \( x = 0 \), \( y = 0 \) into the inequality: \( 0<-\frac{2}{5}(0)+1\), which simplifies to \( 0 < 1 \), a true statement. So the region containing the origin should be shaded.

Now let's analyze the graphs:

• Graph A: The boundary line is solid (since the point \((0,1)\) is a solid dot), but our inequality is strict, so the line should be dashed. Eliminate A.
• Graph B: The boundary line is solid. Eliminate B.
• Graph C: The boundary line is dashed (since the point \((0,1)\) is a hollow dot), and the shading is below the line (containing the origin, as we saw from the test point). Let's check the slope: the line passes through \((0,1)\) and \((5, - 1)\) (since slope \( -\frac{2}{5}=\frac{y_2 - y_1}{x_2 - x_1}\), if \( x_2 = 5 \), \( y_2=1-\frac{2}{5}\times5=1 - 2=-1\)), and the shading is below the line.
• Graph D: The shading is above the line, which would not satisfy \( y < -\frac{2}{5}x + 1 \) (since testing the origin in D's shaded region would be false).

Answer:

C. Graph C