Question

question 21 of 40
which graph represents the solution set to the following system of linear inequalities?
$y < -x + 4$
$y \geq \frac{2}{5}x - 1$
graphs a and b as shown, with text descriptions for graphs

Explanation:

Step1: Analyze the first inequality \( y < -x + 4 \)

The boundary line is \( y=-x + 4 \), which has a slope of \(- 1\) and a y - intercept of \(4\). Since the inequality is \(y < -x + 4\) (strictly less than), the boundary line should be dashed. The region we are interested in is below this line.

Step2: Analyze the second inequality \( y\geq\frac{2}{5}x-1 \)

The boundary line is \( y = \frac{2}{5}x-1\), which has a slope of \(\frac{2}{5}\) and a y - intercept of \(- 1\). Since the inequality is \(y\geq\frac{2}{5}x - 1\) (greater than or equal to), the boundary line should be solid. The region we are interested in is above (or on) this line.

Step3: Analyze the options

• For option A: The dashed line has a slope of \(-1\) (matches \(y=-x + 4\)) and the solid line has a positive slope (matches \(y=\frac{2}{5}x-1\)). The shaded region is above the solid line and below the dashed line, which matches the solution set of the system of inequalities.
• For option B: The shaded region does not seem to be the intersection of the region below \(y=-x + 4\) and above \(y=\frac{2}{5}x-1\). The position of the shaded area and the orientation of the regions relative to the lines do not match the requirements of the inequalities.

Answer:

A