Question

question 5 of 10
which graph shows the solution to this system of inequalities?
$y < -dfrac{1}{3}x + 1$
$y \leq 2x - 3$

graph a is shown with a dashed line for $y = -dfrac{1}{3}x + 1$ and a solid line for $y = 2x - 3$, and a shaded region. option a is a radio button.
text description for graph

Explanation:

Step1: Analyze \( y < -\frac{1}{3}x + 1 \)

The inequality \( y < -\frac{1}{3}x + 1 \) has a dashed line (since it's \( < \), not \( \leq \)) with slope \( -\frac{1}{3} \) and y - intercept 1. The region below this line is shaded.

Step2: Analyze \( y \leq 2x - 3 \)

The inequality \( y \leq 2x - 3 \) has a solid line (since it's \( \leq \)) with slope 2 and y - intercept - 3. The region below or on this line is shaded.

Step3: Find the intersection of the two regions

We need to find the region that is below the dashed line \( y = -\frac{1}{3}x + 1 \) and below or on the solid line \( y = 2x - 3 \). First, find the intersection point of the two lines \( y=-\frac{1}{3}x + 1 \) and \( y = 2x-3 \).

Set \( -\frac{1}{3}x + 1=2x - 3 \)

Multiply through by 3 to eliminate the fraction: \( -x + 3 = 6x-9 \)

Add \( x \) to both sides: \( 3=7x - 9 \)

Add 9 to both sides: \( 12 = 7x \)

So \( x=\frac{12}{7}\approx1.71 \), and \( y = 2\times\frac{12}{7}-3=\frac{24}{7}-\frac{21}{7}=\frac{3}{7}\approx0.43 \)

Now, check the shading: For \( y < -\frac{1}{3}x + 1 \), we shade below the dashed line. For \( y\leq2x - 3 \), we shade below the solid line. The overlapping region (the solution to the system) is the area that satisfies both, which should be below the dashed line and below the solid line.

Looking at the options, in option A, the dashed line has a negative slope (consistent with \( y = -\frac{1}{3}x + 1 \)), the solid line has a positive slope (consistent with \( y = 2x - 3 \)), the dashed line is above the solid line at their intersection, and the shaded region is below both (since it's the overlap of the region below the dashed line and below the solid line).

Answer:

A