Which graph best represents the solution set for this system of inequalities?

To find the solution set for the system of inequalities given by:

1. \( x + 2y < -2 \)
2. \( y - x < 3 \)

We can solve each inequality step by step.

Step 1: Convert inequalities to slope-intercept form (y = mx + b)

For the first inequality: \[ x + 2y < -2 \]

Rearranging it: \[ 2y < -x - 2 \] \[ y < -\frac{1}{2}x - 1 \]

For the second inequality: \[ y - x < 3 \]

Rearranging it: \[ y < x + 3 \]

Step 2: Graph the boundary lines

1. For \( y = -\frac{1}{2}x - 1 \):

   • This is a line with a slope of -1/2 and y-intercept at -1.
   • The line should be dashed since the inequality is "<", indicating that points on the line are not included in the solution.

2. For \( y = x + 3 \):

   • This line has a slope of 1 and a y-intercept at 3.
   • Similarly, this line should also be dashed.

Step 3: Determine the shaded regions

• For \( y < -\frac{1}{2}x - 1 \):

  • Shade the area below the line \( y = -\frac{1}{2}x - 1 \).

• For \( y < x + 3 \):

  • Shade the area below the line \( y = x + 3 \).

Step 4: Identify the intersection of the shaded areas

The solution set for the system of inequalities is the region where the areas shaded for both inequalities overlap.

Conclusion

When graphing these inequalities, you should find a region in the coordinate plane that lies below both dashed lines, which indicates the solution set for the system. You can check various points in this intersection area to ensure they satisfy both original inequalities.

If you have specific graphs provided to reference, look for the graph that shows the region below both lines and ensure it does not include the lines themselves, reflecting the '<' type inequalities.

Which graph best represents the solution set to this system of inequalities?

x + y <1
x - y greater or equal to 2

To find the solution set for the system of inequalities:

1. \( x + y < 1 \)
2. \( x - y \geq 2 \)

We will analyze each inequality.

Step 1: Convert inequalities to slope-intercept form (y = mx + b)

For the first inequality: \[ x + y < 1 \]

Rearranging it: \[ y < -x + 1 \]

This means we will graph the line \( y = -x + 1 \) with a dashed line, since it's a "<" inequality (points on the line are not included in the solution).

For the second inequality: \[ x - y \geq 2 \]

Rearranging it: \[ -y \geq -x + 2 \] \[ y \leq x - 2 \]

This means we will graph the line \( y = x - 2 \) with a solid line, since it's a "greater than or equal to" (≥) inequality (points on the line are included in the solution).

Step 2: Graph the boundary lines

1. Graph \( y = -x + 1 \):

   • This line has a slope of -1 and a y-intercept of 1.
   • Use dashed line for this line.

2. Graph \( y = x - 2 \):

   • This line has a slope of 1 and a y-intercept of -2.
   • Use solid line for this line, as points on the line are included in the solution set.

Step 3: Determine the shaded regions

• For \( y < -x + 1 \):

  • Shade the area below the line \( y = -x + 1 \).

• For \( y \leq x - 2 \):

  • Shade the area below the line \( y = x - 2 \).

Step 4: Identify the intersection of the shaded areas

The solution set for the system of inequalities is where the shaded areas for both inequalities overlap.

Conclusion

1. The first inequality's shaded area will be below the dashed line \( y = -x + 1 \).
2. The second inequality's shaded area will be below the solid line \( y = x - 2 \).

Look for the region on the graph that is below both lines. The intersection of these shaded areas represents the solution set for the system. If you have specific graphs provided to reference, choose the one that shows this overlapping region correctly.

If the graph of the second equation in the system passes through the points (-12, 20) and (4, 12), which statement is true?

Question 10 options:

The system has an infinite number of solutions.

The only solution to the system is (10, 5).

The only solution to the system is (0, 14).

The system has no solutions.

To analyze the system of inequalities given in your earlier question, we have:

1. \( x + y < 1 \)
2. \( x - y \geq 2 \)

You mentioned that the second equation, \( x - y \geq 2 \), passes through the points (-12, 20) and (4, 12). To confirm this and evaluate the situation, let's first find the equation of the line represented by the second inequality using the two points provided.

Step 1: Determine the slope of the line

The slope \( m \) can be calculated using the formula:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Using points (-12, 20) and (4, 12):

\[ m = \frac{12 - 20}{4 - (-12)} = \frac{-8}{16} = -\frac{1}{2} \]

Step 2: Write the equation of the line

Using the point-slope form of the equation:

\[ y - y_1 = m(x - x_1) \]

Taking the point (4, 12):

\[ y - 12 = -\frac{1}{2}(x - 4) \]

Simplifying this:

\[ y - 12 = -\frac{1}{2}x + 2 \ y = -\frac{1}{2}x + 14 \]

Step 3: Express the inequality

We have the line equation \( y = -\frac{1}{2}x + 14 \). The corresponding inequality is \( y \leq -\frac{1}{2}x + 14 \) (we use a solid line, indicating points on the line are included).

Step 4: Analyze the first inequality

The first inequality \( x + y < 1 \) can be rewritten as:

\[ y < -x + 1 \]

Step 5: Analyzing the system of inequalities

Now we can analyze the potential intersections and relationships between the shaded areas:

1. The line for \( x + y < 1 \) has a slope of -1 and a y-intercept of 1.
2. The line for \( x - y \geq 2 \) has a slope of -0.5 and a y-intercept of 14.

Step 6: Determine if the lines are parallel or intersect

The slopes of the two lines are different (-1 and -0.5), indicating that the lines will intersect at some point.

Step 7: Evaluate the options

Given that the two lines intersect but do not represent parallel lines that would lead to either an infinite number of solutions or no solutions, we can conclude:

• There is a solution to the system, as the inequalities can be satisfied at some point within the viable area.
• The system does not have an infinite number of solutions or only single distinct solutions, as the inequalities define a feasible region.

Conclusion

The correct statement would depend on additional analysis or informing conditions of the system. Given the above conclusions, it seems that neither "The only solution to the system is (10, 5)" nor "The only solution to the system is (0, 14)" would be accurate unless further information is available.

Identifying the feasible region suggests:

• The best option among the provided ones would likely be: "The system has an infinite number of solutions."

This is because the area beneath the two boundary lines consists of multiple points that satisfy both inequalities, and not just singular points.