Question

graph the system of linear inequalities.
y ≥ 1
y < 2x + 1

Explanation:

Step1: Analyze \( y \geq 1 \)

The inequality \( y \geq 1 \) represents all points above or on the horizontal line \( y = 1 \). So we look for graphs with a solid or dashed line at \( y = 1 \) and shading above (or on) it. A solid line since the inequality is "greater than or equal to".

Step2: Analyze \( y < 2x + 1 \)

The inequality \( y < 2x + 1 \) represents all points below the line \( y = 2x + 1 \). The line \( y = 2x + 1 \) has a slope of \( 2 \) and a y - intercept of \( 1 \). Since the inequality is "less than", the line should be dashed.

Step3: Match the conditions

We need to find the graph where there is a solid line at \( y = 1 \) (shading above \( y = 1 \)) and a dashed line \( y = 2x + 1 \) (shading below it). Let's check the options:

• For the second graph (the middle one among the four, the one with a horizontal line at \( y = 1 \) (solid) and a dashed line with slope \( 2 \) (since when \( x = 0 \), \( y = 1 \); when \( x = 1 \), \( y = 3 \), so slope \( 2 \)) and shading below the dashed line and above \( y = 1 \). Wait, no, let's re - check. Wait, the line \( y = 2x+1 \): when \( x = 0 \), \( y = 1 \); when \( x = 1 \), \( y=3 \); when \( x=- 1 \), \( y=-1\). The correct graph should have a solid line at \( y = 1 \) (shading above) and a dashed line \( y = 2x + 1 \) (shading below). Let's check the third graph? Wait, no, the second graph (the one with the horizontal line at \( y = 1 \) (solid) and a dashed line going from (0,1) with slope 2 (since from (0,1) to (1,3) is slope 2) and shading below the dashed line and above \( y = 1 \). Wait, actually, let's re - evaluate. The first inequality \( y\geq1 \) is a horizontal line \( y = 1 \), solid, shading above. The second inequality \( y<2x + 1 \): the line \( y = 2x+1 \) has a positive slope. So we need a graph with a solid horizontal line at \( y = 1 \) (shading above) and a dashed line with positive slope (slope 2) and shading below it. Looking at the options, the second graph (the one with the horizontal solid line at \( y = 1 \) and a dashed line \( y = 2x + 1 \) (since when \( x = 0 \), \( y = 1 \); when \( x = 1 \), \( y = 3 \)) and shading below the dashed line and above \( y = 1 \)) matches. Wait, no, maybe the third graph? Wait, no, let's check the slopes. The line \( y=2x + 1 \): slope \( m = 2 \), so it's a steep line. The second graph (the middle one) has a dashed line with slope 2? Wait, when \( x = 0 \), \( y = 1 \); when \( x = 1 \), \( y=3 \), so yes, slope 2. And the solid line at \( y = 1 \) with shading above \( y = 1 \) and below the dashed line. So the correct graph is the second one (the middle graph among the four, the one with the horizontal solid line at \( y = 1 \) and a dashed line \( y = 2x + 1 \) and shading in the region that satisfies both inequalities).

Answer:

The Middle Graph (the second graph from the top among the four given graphs)