Question

find the graph of this system of linear inequalities.$\begin{cases}y < 1 \\y leq 2x - 1end{cases}$

Explanation:

Step1: Analyze $y < 1$

The inequality $y < 1$ represents all points below the dashed horizontal line $y=1$ (dashed because the inequality is strict, no equality).

Step2: Analyze $y \leq 2x - 1$

First, identify the boundary line $y=2x-1$: this is a solid line (since the inequality includes equality) with slope $2$ and y-intercept $-1$. The inequality $y \leq 2x - 1$ represents all points below or on this solid line.

Step3: Find overlapping region

The solution is the area that satisfies both inequalities: below $y=1$ (dashed line) and below/on $y=2x-1$ (solid line). This matches the first graph, where the purple region is the overlap of the area below $y=1$ (blue) and below $y=2x-1$ (red).

Answer:

The correct graph is the first (leftmost) grid, with the purple overlapping region below the dashed line $y=1$ and below/on the solid line $y=2x-1$.