Consider the following system of inequalities.

Question

y≥2x-6
y<5x+2
is (5,2) a solution to the system of inequalities? explain.
1) graph the system of inequalities
2) when looking at the graph where is the solution set to the system of inequalities
3) could a system of linear inequalities ever have no solution? what would this graph look like?

Answer 1

Let's analyze the system of inequalities step by step:

Given Inequalities:

1. \( y \geq 2x - 6 \)
2. \( y < 5x + 2 \)

Step 1: Checking if (5, 2) is a solution

To determine if the point \( (5, 2) \) is a solution to the system, we can substitute \( x = 5 \) and \( y = 2 \) into each inequality:

1. For \( y \geq 2x - 6 \): \[ 2 \geq 2(5) - 6 \] \[ 2 \geq 10 - 6 \] \[ 2 \geq 4 \quad \text{(False)} \]

2. For \( y < 5x + 2 \): \[ 2 < 5(5) + 2 \] \[ 2 < 25 + 2 \] \[ 2 < 27 \quad \text{(True)} \]

Since the first inequality is false and the second one is true, the point \( (5, 2) \) is not a solution to the system.

Step 2: Graphing the system of inequalities

Graphing Inequality 1: \( y \geq 2x - 6 \)

• Start with the line \( y = 2x - 6 \). This line has a slope of 2 and a y-intercept of -6.
• Plot the line and shade above it (since it is \( \geq \)).

Graphing Inequality 2: \( y < 5x + 2 \)

• Start with the line \( y = 5x + 2 \). This line has a slope of 5 and a y-intercept of 2.
• Plot the line and shade below it (since it is \( < \)).

Step 3: Finding the solution set

The solution set for the system of inequalities is where the shaded regions of both inequalities overlap. This area represents all the points \( (x, y) \) that satisfy both inequalities.

Step 4: System of linear inequalities could have no solution

Yes, a system of linear inequalities can indeed have no solution. For example, if the graphs of the inequalities are parallel lines without any points in common, there will be no region that satisfies both inequalities.

Example:

1. \( y < 2x + 1 \) (a line with a slope of 2)
2. \( y \geq 2x + 3 \) (also a line with a slope of 2)

The first line would be shaded below it, and the second line would be shaded above it. Because these lines are parallel and do not intersect, there would be no common shaded area - hence, no solution.

Summary

1. The point \( (5, 2) \) is not a solution.
2. The solution set is the overlapping shaded region from both inequalities.
3. Systems of linear inequalities can have no solutions; this occurs when the lines are parallel and do not intersect.