Question

which graph shows the solution to the system of linear inequalities?
x - 4y ≤ 4
y < x + 1

Explanation:

Step1: Rewrite first inequality to slope-intercept

Rearrange $x-4y\leq4$ to solve for $y$:

$$\begin{align*} -4y&\leq -x + 4\\ y&\geq \frac{1}{4}x - 1 \end{align*}$$

This means a solid line (due to $\leq$) and shading above the line.

Step2: Rewrite second inequality to slope-intercept

The inequality $y < x + 1$ is already in slope-intercept form:
$$y < x + 1$$
This means a dashed line (due to $<$) and shading below the line.

Step3: Identify overlapping region

The solution is the area that is shaded above the solid line $y=\frac{1}{4}x - 1$ and below the dashed line $y=x+1$.

Answer:

The first graph (leftmost option, with overlapping purple shading in the upper right region, solid lower line, dashed upper line) is the correct solution.