Question

On a coordinate plane, 2 lines are shown. The first dashed straight line has a positive slope and goes through (negative 1, 0) and (0, 2). Everything to the right of the line is shaded. The second solid straight line has a positive slope and goes through (0, negative 1) and (1, 1). Everything to the left of the line is shaded. Which equation represents an inequality in the system of inequalities shown in the graph? Which point is a solution to the system?

Answer 1

The equation that represents an inequality in the system is y ≤ -2x + 2. The point that is a solution to the system is (0, -1).

Equation for the Inequality:

Let's denote the first dashed line as Line 1 and the second solid line as Line 2. The inequality for the system can be represented as follows:

`\[ \text{Equation for Line 1:} \quad y - y_1 = m_1(x - x_1) \]`

`\[ \text{Equation for Line 2:} \quad y - y_2 = m_2(x - x_2) \]`

Given that Line 1 passes through \((-1, 0)\) and \((0, 2)\) with a positive slope, we can find its equation. Similarly, Line 2 passes through \((0, -1)\) and \((1, 1)\) with a positive slope.

Let's find the equations for Line 1 and Line 2 and then write an inequality representing the system.

Solution Point:

To find the solution point, you need to identify the coordinates where the shaded regions overlap. This point will satisfy both inequalities.

Let me calculate the equations and find the solution point.

Equation for Line 1:

Using the points \((-1, 0)\) and \((0, 2)\), we find the slope (\(m_1\)):

`\[ m_1 = (2 - 0)/(0 - (-1)) = (2)/(1) = 2 \]`

Now, using the point \((-1, 0)\):

`\[ y - 0 = 2(x - (-1)) \]`

`\[ y = 2x + 2 \]`

Equation for Line 2:

Using the points \((0, -1)\) and \((1, 1)\), we find the slope (\(m_2\)):

`\[ m_2 = (1 - (-1))/(1 - 0) = (2)/(1) = 2 \]`

Now, using the point \((0, -1)\):

`\[ y - (-1) = 2(x - 0) \]`

`\[ y = 2x - 1 \]`

Inequality:

The equation that represents an inequality in the system is y ≤ -2x + 2.

Points satisfying the system:

A point is considered a solution to the system if it satisfies both inequalities. We need to analyze each option:

Option 1: (0, 0)

Substituting this point into both inequalities:

0 ≤ -2(0) + 2 (true)

0 ≥ 2(0) - 1 (false)

Therefore, (0, 0) is not a solution.

Option 2: (1, 1)

Substituting this point into both inequalities:

1 ≤ -2(1) + 2 (false)

1 ≥ 2(1) - 1 (true)

Therefore, (1, 1) is not a solution.

Option 3: (0, -1)

Substituting this point into both inequalities:

-1 ≤ -2(0) + 2 (true)

-1 ≥ 2(0) - 1 (true)

Therefore, (0, -1) is a solution.

Answer 2

The equations representing the inequalities are y > 2x + 2 for the first line and y < 2x - 1 for the second line. To find a solution to the system of inequalities, we need a point that lies within the shared shaded region but without its specific coordinates, we can't pinpoint an exact solution.

• To represent the inequality for the first line on a coordinate plane, we can begin by finding the equation of the line using the two points provided, which are (negative 1, 0) and (0, 2).
• To find the slope (m) of the line, we use the formula Δy/Δx which gives us (2 - 0)/(0 - (-1)) = 2/1 = 2.
• Knowing that the y-intercept (b) is 2, from the point (0, 2), the equation of the line is y = 2x + 2.
• Since the area to the right of this dashed line is shaded, the inequality representing this situation would be y > 2x + 2.
• Similarly, for the second solid line, we use points (0, negative 1) and (1, 1).
• Calculating the slope gives us (1 - (-1))/(1 - 0) = 2/1 = 2 as well.
• With the y-intercept being -1, from the point (0, -1), the equation is y = 2x - 1.
• As the area to the left is shaded, the corresponding inequality would be y < 2x - 1.
• To find a point that is a solution to the system of inequalities, we can look for a point that satisfies both conditions.
• For instance, the point (0, 0) fulfills both y > 2x + 2 (since 0 is not greater than 2) and y < 2x - 1 (since 0 is less than -1), therefore (0, 0) would not be a solution.
• However, the point (0, 1) satisfies y > 2(0) + 2 (since 1 is not greater than 2) and y < 2(0) - 1 (since 1 is greater than -1), hence, (0, 1) is not a valid solution either.
• A valid solution would be a point that lies within the area that is shaded by both inequalities.
• Identifying such a point on the graph would be required, but based on the instructions above, it is not possible to visually determine it.
• Nevertheless, we know that it would have to satisfy y > 2x + 2 and y < 2x - 1 simultaneously.