Read: Define Solutions to Systems of Linear Inequalities
Learning Objectives
- Graph a system of linear inequalities and define the solutions region
- Verify whether a point is a solution to a system of inequalities
- Identify when a system of inequalities has no solution
Graph a system of two inequalities
Remember from the module on graphing that the graph of a single linear inequality splits the coordinate plane into two regions. On one side lie all the solutions to the inequality. On the other side, there are no solutions. Consider the graph of the inequality [latex]y<2x+5[/latex].![An upward-sloping dotted line with the region below it shaded. The shaded region is labeled y is less than 2x+5. A is equal to (-1,1). B is equal to (3,1).](https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/1468/2016/02/04064400/image012.gif)
![Downward-sloping dotted line with the region above it shaded. The shaded region is y is greater than negative x. Point M=(-2,3). Point N=(4,-1).](https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/1468/2016/02/04064401/image013.gif)
![The two previous graphs combined. A blue dotted line with the region above shaded and labeled y is greater than negative x. A red dotted line with the region below it shaded and labeled y is less than 2x+5. The region where the shaded areas overlap is labeled y is greater than negative x and y is less than 2x+5. The point M equals (-2,3) and is in the blue shaded region. The point A equals (-1,-1) and is in the red shaded region. The point B equals (3,1) and is in the purple overlapping region. The point N equals (4,-1) and is also in the purple overlapping region.](https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/1468/2016/02/04064403/image014.gif)
Determine whether an ordered pair is a solution to a system of linear inequalities
![image014](https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/1468/2016/02/04064403/image014-300x297.gif)
Example
Is the point [latex](2, 1)[/latex] a solution of the system [latex]x+y>1[/latex] and [latex]2x+y<8[/latex]?Answer: Check the point with each of the inequalities. Substitute [latex]2[/latex]for x and [latex]1[/latex] for y. Is the point a solution of both inequalities? [latex-display]\begin{array}{r}x+y>1\\2+1>1\\3>1\\\text{TRUE}\end{array}[/latex-display] [latex](2, 1)[/latex] is a solution for [latex]x+y>1[/latex]. [latex-display]\begin{array}{r}2x+y<8\\2\left(2\right)+1<8\\4+1<8\\5<8\\\text{TRUE}\end{array}[/latex-display] [latex-display](2, 1)[/latex] is a solution for [latex]2x+y<8.[/latex-display] Since [latex](2, 1)[/latex] is a solution of each inequality, it is also a solution of the system.
Answer
The point [latex](2, 1)[/latex] is a solution of the system [latex]x+y>1[/latex] and [latex]2x+y<8[/latex].![Two dotted lines, one red and one blue. The region below the blue dotted line is shaded and labeled 2x+y is less than 8. The region above the dotted red line is shaded and labeled x+y is greater than 1. The overlapping shaded region is purple and is labeled x+y is greater than 1 and 2x+y is less than 8. The point (2,1) is in the overlapping purple region.](https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/1468/2016/02/04064404/image015.gif)
Example
Is the point [latex](2, 1)[/latex] a solution of the system [latex]x+y>1[/latex] and [latex]3x+y<4[/latex]?Answer: Check the point with each of the inequalities. Substitute [latex]2[/latex] for x and [latex]1[/latex] for y. Is the point a solution of both inequalities? [latex-display]\begin{array}{r}x+y>1\\2+1>1\\3>1\\\text{TRUE}\end{array}[/latex-display] [latex](2, 1)[/latex] is a solution for [latex]x+y>1[/latex]. [latex-display]\begin{array}{r}3x+y<4\\3\left(2\right)+1<4\\6+1<4\\7<4\\\text{FALSE}\end{array}[/latex-display] [latex](2, 1)[/latex] is not a solution for [latex]3x+y<4[/latex]. Since [latex](2, 1)[/latex] is not a solution of one of the inequalities, it is not a solution of the system.
Answer
The point [latex](2, 1)[/latex] is not a solution of the system [latex]x+y>1[/latex] and [latex]3x+y<4[/latex].![A downward-sloping bue dotted line with the region below shaded and labeled 3x+y is less than 4. A downward-sloping red dotted line with the region above it shaded and labeled x+y is greater than 1. An overlapping purple shaded region is labeled x+y is greater than 1 and 3x+y is less than 4. A point (2,1) is in the red shaded region, but not the blue or overlapping purple shaded region.](https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/1468/2016/02/04064405/image016.gif)
- Graph each inequality as a line and determine whether it will be solid or dashed
- Determine which side of each boundary line represents solutions to the inequality by testing a point on each side
- Shade the region that represents solutions for both inequalities
Systems with no solutions
In the next example, we will show the solution to a system of two inequalities whose boundary lines are parallel to each other. When the graphs of a system of two linear equations are parallel to each other, we found that there was no solution to the system. We will get a similar result for the following system of linear inequalities.Examples
Graph the system [latex]\begin{array}{c}y\ge2x+1\\y\lt2x-3\end{array}[/latex]Answer: The boundary lines for this system are parallel to each other, note how they have the same slopes.
[latex]\begin{array}{c}y=2x+1\\y=2x-3\end{array}[/latex]
Plotting the boundary lines will give the graph below. Note that the inequality [latex]y\lt2x-3[/latex] requires that we draw a dashed line, while the inequality [latex]y\ge2x+1[/latex] will require a solid line.![y=2x+1](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/121/2016/06/01183205/Screen-Shot-2016-05-13-at-1.56.45-PM-300x300.png)
[latex]\begin{array}{c}y\ge2x+1\\0\ge2\left(0\right)+1\\0\ge{1}\end{array}[/latex]
This is not true, so we know that we need to shade the other side of the boundary line for the inequality [latex]y\ge2x+1[/latex]. The graph will now look like this:![y=2x+1](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/121/2016/06/01183206/Screen-Shot-2016-05-13-at-2.02.49-PM-300x300.png)
[latex]\begin{array}{c}y\lt2x-3\\0\lt2\left(0,\right)x-3\\0\lt{-3}\end{array}[/latex]
This is not true, so we know that we need to shade the other side of the boundary line for the inequality[latex]y\lt2x-3[/latex]. The graph will now look like this:![y=2x+1](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/121/2016/06/01183208/Screen-Shot-2016-05-13-at-2.07.01-PM-297x300.png)
Summary
- Solutions to systems of linear inequalities are entire regions of points
- You can verify whether a point is a solution to a system of linear inequalities in the same way you verify whether a point is a solution to a system of equations
- Systems of inequalities can have no solutions when boundary lines are parallel
Licenses & Attributions
CC licensed content, Shared previously
- Ex 1: Graph a System of Linear Inequalities. Authored by: James Sousa (Mathispower4u.com) . License: CC BY: Attribution.
- Ex 2: Graph a System of Linear Inequalities. Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution.
- Unit 14: Systems of Equations and Inequalities, from Developmental Math: An Open Program. Provided by: Monterey Institute of Technology Located at: https://www.nroc.org/. License: CC BY: Attribution.