Question

Write a system of linear inequalities represented by the graph.

A system of linear inequalities is drawn on a coordinate plane. The x-axis ranges from negative 4 to 4, in increments of 1. The y-axis ranges from negative 3 to 5, in increments of 1. A dashed line passes through ordered pair negative 3 comma 3, ordered pair negative 1 comma negative 1, and ordered pair 0 comma negative 3. Another dashed line passes through ordered pair negative 2 comma 3, ordered pair negative 1 comma 1, and ordered pair 1 comma negative 3.

Answer 1

Explanation:

To write a system of linear inequalities represented by the given graph, we need to analyze the lines and their slopes.

Let's start with the dashed line passing through the ordered pairs (-3, 3), (-1, -1), and (0, -3). Since this line is dashed, it indicates that the inequality is strict.

1. Find the slope of the line using the formula: slope = (change in y) / (change in x).

The slope of the line passing through (-3, 3) and (-1, -1) is: (3 - (-1)) / (-3 - (-1)) = 4 / -2 = -2.

2. Write the equation of the line using the point-slope form: y - y₁ = m(x - x₁), where (x₁, y₁) is any point on the line.

Using the point (-3, 3) and the slope -2, we have the equation: y - 3 = -2(x - (-3)).

Simplifying the equation, we get: y - 3 = -2(x + 3).

3. Rearrange the equation to isolate y: y = -2x - 3.

Now, let's move on to the dashed line passing through the ordered pairs (-2, 3), (-1, 1), and (1, -3). Again, since this line is dashed, it indicates that the inequality is strict.

1. Find the slope of the line using the formula: slope = (change in y) / (change in x).

The slope of the line passing through (-2, 3) and (-1, 1) is: (3 - 1) / (-2 - (-1)) = 2 / -1 = -2.

2. Write the equation of the line using the point-slope form: y - y₁ = m(x - x₁), where (x₁, y₁) is any point on the line.

Using the point (-2, 3) and the slope -2, we have the equation: y - 3 = -2(x - (-2)).

Simplifying the equation, we get: y - 3 = -2(x + 2).

3. Rearrange the equation to isolate y: y = -2x - 1.

Therefore, the system of linear inequalities represented by the graph is:

y < -2x - 3 (dashed line passing through (-3, 3), (-1, -1), and (0, -3))

y < -2x - 1 (dashed line passing through (-2, 3), (-1, 1), and (1, -3))

Answer 2

The equations of the dashed lines are y = 2x + 9 and y = 2x + 5.

To write a system of linear inequalities represented by the graph, we can start by identifying the equations of the two dashed lines.

Let's call the first line L1 and the second line L2.

For L1, we can find the equation using the points negative 3 comma 3 and negative 1 comma negative 1.

The slope of L1 is (3 - (-1))/(negative 3 - (-1)) = 4/2 = 2. So the equation of L1 is y = 2x + b.

We can substitute one of the points into this equation to find the value of b.

Using the point negative 3 comma 3, we get 3 = 2 * (-3) + b, which simplifies to b = 9. Therefore, the equation of L1 is y = 2x + 9.

We can use the same process to find the equation of L2.

Using the points negative 2 comma 3 and negative 1 comma 1, we get the equation y = 2x + 5.

Finally, we can write the system of linear inequalities as:

y > 2x + 9

y > 2x + 5

This represents the shaded region above both dashed lines on the graph.