Question

write a system of linear inequalities represented by the graph. inequality 1: inequality 2:

Explanation:

Step1: Identify the vertical lines

The graph has two vertical dashed lines. One at \( x = 2 \) (wait, looking at the grid, the first dashed line is at \( x = 2 \)? Wait, the grid has x-axis with -1,1,3,5. Wait, the blue region is between two vertical dashed lines. Let's check the x-values. The left dashed line: looking at the grid, the first dashed line (left) is at \( x = 2 \)? Wait, no, the grid lines: from x=-1, 1, 3, 5. Wait, the blue region is between two vertical dashed lines. Let's see the x-coordinates. The left dashed line: let's count the grid squares. From x=1 to x=3, the first dashed line is at x=2? Wait, no, maybe the left line is \( x = 2 \) and the right line is \( x = 4 \)? Wait, no, looking at the graph, the two vertical dashed lines: one is at \( x = 2 \) (wait, no, the grid has x=1, 3, so between x=2? Wait, maybe the left line is \( x > 2 \) and the right line is \( x < 4 \)? Wait, no, the graph shows the blue region between two vertical dashed lines. Let's re-examine: the x-axis has marks at -1, 1, 3, 5. The left dashed line is at x=2 (between 1 and 3), and the right dashed line is at x=4 (between 3 and 5). Wait, no, maybe the left line is \( x = 2 \) (dashed) and the right line is \( x = 4 \) (dashed). Wait, but the problem is to write the system. Wait, maybe the left line is \( x > 2 \) (since the blue is to the right of the left dashed line) and the right line is \( x < 4 \) (since the blue is to the left of the right dashed line). Wait, no, looking at the graph, the two vertical dashed lines: let's check the x-values. The left dashed line: when x=2 (since from x=1 to x=3, the middle is x=2), and the right dashed line is at x=4 (from x=3 to x=5, middle is x=4). Wait, but maybe the lines are \( x > 2 \) and \( x < 4 \)? Wait, no, maybe the left line is \( x = 2 \) (dashed) so the inequality is \( x > 2 \) (since the blue is to the right of the left dashed line) and the right line is \( x = 4 \) (dashed) so the inequality is \( x < 4 \) (since the blue is to the left of the right dashed line). Wait, but maybe I made a mistake. Alternatively, maybe the left line is \( x > 2 \) and the right line is \( x < 4 \). Wait, but let's think again. Vertical lines: the equation of a vertical line is \( x = a \). If the line is dashed, the inequality is strict (>, <). The blue region is between the two lines, so for the left line (dashed), the region is to the right of the line, so \( x > 2 \), and for the right line (dashed), the region is to the left of the line, so \( x < 4 \). Wait, but maybe the lines are \( x > 2 \) and \( x < 4 \). Alternatively, maybe the left line is \( x = 2 \) (dashed) so \( x > 2 \), and the right line is \( x = 4 \) (dashed) so \( x < 4 \). So the system is \( x > 2 \) and \( x < 4 \). Wait, but let's check the graph again. The blue region is between two vertical dashed lines. So the left inequality: the blue is to the right of the left dashed line, so \( x > 2 \) (if the left line is \( x = 2 \)), and the right inequality: the blue is to the left of the right dashed line, so \( x < 4 \) (if the right line is \( x = 4 \)). Alternatively, maybe the lines are \( x > 2 \) and \( x < 4 \). Wait, but maybe the left line is \( x = 2 \) (dashed) so \( x > 2 \) and the right line is \( x = 4 \) (dashed) so \( x < 4 \). So the system is \( x > 2 \) and \( x < 4 \). Wait, but maybe I misread the graph. Let's assume that the two vertical dashed lines are at \( x = 2 \) and \( x = 4 \). So the inequalities are \( x > 2 \) (since the blue is to the right of \( x = 2 \), dashed line means strict inequality) and \( x < 4 \) (since the blue is to the left of \( x = 4 \), dashed line means strict inequality).

Step2: Write the inequalities

For the left dashed line (x=2, dashed), the region is to the right, so \( x > 2 \). For the right dashed line (x=4, dashed), the region is to the left, so \( x < 4 \).

Answer:

Inequality 1: \( x > 2 \)
Inequality 2: \( x < 4 \)

Wait, but maybe the lines are \( x = 2 \) and \( x = 4 \), so the system is \( x > 2 \) and \( x < 4 \). Alternatively, maybe the left line is \( x = 2 \) (dashed) so \( x > 2 \) and the right line is \( x = 4 \) (dashed) so \( x < 4 \). So the two inequalities are \( x > 2 \) and \( x < 4 \).