Question

which system of linear inequalities is represented by the graph?
y ≥ x − 2 and y ≤ x + 1
y < x − 2 and y > x + 1
y ≤ x − 2 and y ≥ x + 1
y > x − 2 and y < x + 1

Explanation:

Step1: Analyze the lines' slopes and intercepts

The two lines in the graph have a slope of 1 (since the equations are in the form \(y = x + b\)). For the line \(y = x + 1\), the y - intercept is 1. For the line \(y=x - 2\), the y - intercept is - 2.

Step2: Analyze the inequality signs based on shading

• For the line \(y=x + 1\): The shaded region is below this line. Since the line is dashed (we can infer from the fact that the overlapping region's boundaries are dashed as the inequalities are strict or non - strict? Wait, looking at the options, the lines are either with \(>\), \(<\), \(\geq\), \(\leq\). The shading for \(y=x + 1\): the region that is part of the solution is below the line, and if we check the options, the inequality should be \(y
• For the line \(y=x - 2\): The shaded region is above this line. So the inequality for this line is \(y>x - 2\).

Let's check each option:

• Option 1: \(y\geq x - 2\) and \(y\leq x + 1\): The shading would be between the lines but including the lines. But in the graph, the lines seem to be dashed (since the inequalities in the correct option are strict), so this is wrong.
• Option 2: \(yx + 1\): The regions \(yx + 1\) (above \(y=x + 1\)) do not overlap, so this is wrong.
• Option 3: \(y\leq x - 2\) and \(y\geq x + 1\): The regions \(y\leq x - 2\) (below or on \(y=x - 2\)) and \(y\geq x + 1\) (above or on \(y=x + 1\)) do not overlap, so this is wrong.
• Option 4: \(y>x - 2\) (above \(y=x - 2\)) and \(y

Answer:

\(y > x - 2\) and \(y < x + 1\) (the fourth option: \(y>x - 2\) and \(y