Graphing Systems of Inequalities

Learning Objectives

Introduction

A solution of a system of linear equalities is any ordered pair that is true for all of the equations in the system. Likewise, a solution of a system of linear inequalitiesTwo or more linear inequalities with the same variables. is any ordered pair that is a solution for all of the inequalities in the system.

Graphs are used to show all of the values that are solutions for a system of linear inequalities.

Graphing a System of Two Inequalities

The graph of a single linear inequality splits the coordinate plane A plane formed by the intersection of a horizontal number line called the `x`-axis and a vertical number line called the `y`-axis. 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 `y<2x + 5`.

A graph shows a coordinate plane. A dashed line has a positive slope and crosses the x-axis at (negative 2.5, zero) and heads towards the y-axis. The area below and to the right of the line is shaded. The inequality, y is less than 2 x plus 5, is shown in the shaded area. A point labeled A is shown at (negative 1, negative 1). A point labeled B is shown at (3, 1).

The dashed line is `y = 2x + 5`. Every ordered pair in the colored area below the line is a solution to `y<2x + 5`, as all of the points below the line will make the inequality true. If you doubt that, try substituting the `x` and `y` coordinates of points A and B into the inequality—you’ll see that they work. So, the shaded area shows all of the solutions for this inequality.

The boundary lineA line that divides the coordinate plane into two regions. If points along the boundary line are included in the solution set, then a solid line is used; if points along the boundary line are not included then a dotted line is used. divides the plane into two half planes. In this case, it is shown as a dashed line, as the points on the line don’t satisfy the inequality. If the inequality had been `y<=2x + 5`, then the boundary line would have been solid.

Let’s graph another inequality: `y > -x`. You can check a couple of points to determine which side of the boundary line to shade. Checking points M and N yields true statements. So, we shade the area above the line. The line is dashed, as points on the line don't satisfy the inequality.

A graph shows a coordinate plane. A dashed line with a negative slope crosses the origin. The area above and to the right of the line is shaded. The inequality y is greater than negative x is shown in the shaded area. A point labeled M is shown at (negative 2, 3). A point labeled N is shown at (4, negative 1).

To create a system of inequalities, you need to graph two or more inequalities together. Let’s use `y<2x + 5` and `y> -x` since we have already graphed each of them.

A graph shows a coordinate plane. The plane contains two dashed lines that intersect, 3 shaded areas, inequalities, and points. A dashed line with a negative slope passes through the origin at (0, 0) and the area to the right is shaded. A dashed line with a positive slope passes through the x-axis at about (negative 2.5, 0) and the area to the right is shaded. Shaded sections that overlap are in quadrant 1, part of quadrant 2, and part of quadrant 4. The shaded area to the right of the negative slope shows the inequality, y is greater than negative x, and the point, M equals (negative 2, 3). In the overlapping shaded area, it also shows the inequality, y is greater than negative x and y is less than 2 x plus 5, and two points are labeled, B equals (3, 1), and N equals (4, negative 1). The shaded area to the right of the positive slope overlaps with the shaded area containing points B and N. The shaded area also has the inequality, y is less than 2 x plus 5, and a point is labeled A equals (negative 1, negative 1).

The purple area shows where the solutions of the two inequalities overlap. This area is the solution to the system of inequalities. Any point within this purple region will be true for both `y> -x` and `y<2x + 5`.

On the graph, you can see that the points B and N are solutions for the system because their coordinates will make both inequalities true statements.

In contrast, points M and A both lie outside the solution region (purple). While point M is a solution for the inequality `y> -x` and point A is a solution for the inequality `y<2x + 5`, neither point is a solution for the system.

Example

Problem

Is the point `(2, 1)` a solution of the system `x + y>1` and `2x + y<8` ?

`x + y>1` 

`2 + 1>1` 

`3>1`

 

 

TRUE

`(2, 1)` is a solution for `x + y>1`.

`2x + y<8` 

`2(2) + 1<8` 

`4 + 1<8` 

`5<8` 

 

TRUE

`(2, 1)` is a solution for `2x + y<8`.

Check the point with each of the inequalities. Substitute `2` for `x` and `1` for `y`. Is the point a solution of both inequalities?

 

Answer

The point `(2, 1)` is a solution of the system `x + y>1` and `2x + y<8`.

Since `(2, 1)` is a solution of each inequality, it is also a solution of the system.

Here is a graph of the system in the example above. Notice that `(2, 1)` lies in the purple area, which is the overlapping area for the two inequalities.

A graph shows a coordinate plane. The plane contains two dashed lines, both with negative slopes, that intersect. One dashed line crosses the y-axis at (0, 1) and the x-axis at (1, 0). The area to the right is shaded and shows the inequality, x plus y is greater than 1. The other dashed line crosses the y-axis at (0, 8) and the x-axis at (4, 0). The area to the left is shaded. The inequality, 2 x plus y is less than 8. The overlapping shaded area contains the inequalities, x plus y is greater than 1 and 2 x plus y is less than 8, and shows the point, (2, 1).

Example

Problem

Is the point `(2, 1)` a solution of the system `x + y>1` and `3x + y<4` ?

 

`x + y>1`

`2 + 1>1`

`3>1`

 

 

TRUE

`(2, 1)` is a solution for `x + y>1`.

`3x + y<4` 

`3(2) + 1<4` 

`6 + 1<4` 

`7<4` 

 

FALSE

`(2, 1)` is not a solution for `3x + y<4`.

Check the point with each of the inequalities. Substitute `2` for `x` and `1` for `y`. Is the point a solution of both inequalities?

 

 

 

 

 

Answer

  

The point `(2,1)` is not a solution of the system. 

 `x + y>1` and `3x + y<4`. 

Since `(2, 1)` is not a solution of one of the inequalities, it is not a solution of the system.

Here is a graph of this system. Notice that `(2, 1)` is not in the purple area, which is the overlapping area; it is a solution for one inequality (the red region), but it is not a solution for the second inequality (the blue region).

A graph shows a coordinate plane. The plane contains two dashed lines, both with negative slopes, that intersect. One dashed line crosses the y-axis at (0, 1) and the x-axis at (1, 0). The area to the right is shaded and shows the inequality, x plus y is greater than 1, and the point, (2, 1). The other dashed line crosses the y-axis at (0, 4) and the x-axis at (1.5, 0). The area to the left is shaded. The inequality, 3 x plus y is less than 4. The overlapping shaded area contains the inequalities, x plus y is greater than 1 and 3 x plus y is less than 4.

Which of the points listed below are solutions for the system?

`y>x` 

`x - 2<0` 

 

I. `(1, 1)`

II. `(-5, 9)`

III. `(0, 7)`

 

A) I and II

 

B) II and III

 

C) I and III

 

D) II only

 

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Solving Systems of Inequalities by Graphing

As shown above, finding the solutions of a system of inequalities can be done by graphing each inequality and identifying the region they share.

Example

Problem

Find the solution to the system `x + y>=1` and `y - x>=5` .

A graph shows a coordinate plane. A straight line with a negative slope is labeled with the inequality x plus y is greater than or equal to 1.

Graph one inequality. First, graph the boundary line using a table of values, intercepts, or any other method you prefer. The boundary line for `x + y>=1` is `x + y = 1`, or `y = -x + 1`. Since the equal sign is included with the greater than sign, the boundary line is solid.

 

Test `1`: `(-3, 0)`

`x + y>=1`

`-3 + 0>=1`

`-3>=1`

FALSE

 

Test `2`: `(4, 1)`

`x + y>=1`

`4 + 1>=1`

`5>=1`

TRUE

 

Find an ordered pair on either side of the boundary line. Insert the `x`- and `y`-values into the inequality `x + y>=1` and see which ordered pair results in a true statement.

 

A graph shows a coordinate plane. A solid line with a negative slope crosses the y-axis at (zero, 1) and the x-axis at (1, zero). The area above and to the right of the line is shaded. The inequality x plus y is greater than or equal to 1 is shown in the shaded area. The point (4, 1) is shown in the shaded area. The point (negative 3, 0) is shown in the non-shaded area.

Since `(4, 1)` results in a true statement, the region that includes `(4, 1)` should be shaded.

 

 

A graph shows a coordinate plane. The plane contains two dashed lines, both with negative slopes, that intersect. One solid line has a negative slope and crosses the y-axis at (0, 1) and the x-axis at (1, 0). The area to the right is shaded and shows the inequality, x plus y is greater than or equal to 1, and the point, (0, 6). The other line has a positive slope and crosses the x-axis at (negative 5, 0) and the y-axis at (0, 5) The area to the left is shaded and shows the inequality, y minus x is greater than or equal to 5. The overlapping shaded area contains the point, (0, 6). A point (negative 3, 0) is in the non-shaded area.

Do the same with the second inequality. Graph the boundary line, then test points to find which region is the solution to the inequality. In this case, the boundary line is `y - x = 5` (or `y = x + 5`) and is solid. Test point `(-3, 0)` is not a solution of `y - x>=5`, and test point `(0, 6)` is a solution.

Answer

 

 

 

 

 

 

 

 

The purple region in this graph shows the set of all solutions of the system.

A graph of a coordinate plane shows two solid lines. The first solid line with a negative slope crosses the y-axis at (zero, 1) and the x-axis at (1, zero), and is labeled x plus y is greater than or equal to 1. The second solid line with a positive slope crosses the x-axis at (negative 5, 0) and the y-axis at (zero, 5), and is labeled y minus x is greater than or equal to 5. The lines intersect at (negative 2, 3). The area above the intersection overlaps and is shaded, and contains both the inequalities, x plus y is greater than 1 and y minus x is greater than or equal to 5.
       

 

Example

Problem

Find the solution to the system `3x + 2y<12`  and `-1<=y<=5` .

A coordinate plane shows a dashed line with a negative slope. The line crosses the y-axis at (zero, 6) and the x-axis at (4, zero). The area below and to the left of the line is shaded. The inequality 3 x plus 2 y is less than 12 is shown.

Graph one inequality. First, graph the boundary line, then test points.

 

Remember, because the inequality `3x + 2y<12` does not include the equal sign, draw a dashed border line.

 

Testing a point such as `(0, 0)` will show that the area below the line is the solution to this inequality.

A coordinate plane shows a dashed line with a negative slope. The line crosses the y-axis at (zero, 6) and the x-axis at (4, zero). The area below and to the left of the line is shaded, and in the shaded area the inequality 3 x plus 2 y is less than 12 is shown. The plane also shows two solid horizontal lines. One horizontal line crosses the y-axis at 5. The area below it is shaded. The next horizontal line crosses the y-axis at negative 1. The area above it is shaded. The area between the two horizontal lines contains the inequality, negative 1 is less than or equal to y is less than or equal to 5. The area left of the dashed line and between the two solid lines overlap.

 

The inequality `-1<=y<=5` is actually two inequalities: `-1<=y`, and `y<=5`. Another way to think of this is that `y` must be between `-1` and `5`. The border lines for both are horizontal. The region between those two lines contains the solutions of `-1<=y<=5`. We make the lines solid because we also want to include `y = -1` and `y = 5`.

Graph this region on the same axes as the other inequality.

Answer

 

 

 

 

 

 

 

 

The purple region in this graph shows the set of all solutions of the system.

A coordinate plane shows a dashed line with a negative slope. The line crosses the y-axis at (zero, 6) and the x-axis at (4, zero). The plane also shows two solid horizontal lines. One horizontal line crosses the y-axis at 5. The next horizontal line crosses the y-axis at negative 1. The area between the two horizontal lines contains two inequalities. Negative 1 is less than or equal to y is less than or equal to 5 and 3 x plus 2 y is less than 12. The area left of the dashed line and between the two solid lines overlaps and is shaded.

 

In which of the following is the purple region the solution for the system?

`y>x`

`y<-3x+6`

 

A)

A coordinate plane shows two dashed lines that intersect, with 3 shaded areas. A dashed line with a negative slope passes the y-axis at (0, 6) and the x-axis at (2, 0) and the area to the left is shaded. A dashed line with a positive slope passes through the origin (0, 0) and the area to the left is shaded. Shaded sections that overlap are a little in quadrant 1, most of quadrant 2, and half of quadrant 3.

 

B)

A coordinate plane shows two solid lines that intersect, with 3 shaded areas. A solid line with a negative slope passes the y-axis at (0, 6) and the x-axis at (2, 0) and the area to the left is shaded. A solid line with a positive slope passes through the origin (0, 0) and the area to the left is shaded. Shaded sections that overlap are a little in quadrant 1, most of quadrant 2, and half of quadrant 3.

 

C)

A coordinate plane shows two solid lines that intersect, with 3 shaded areas. A solid line with a negative slope passes the y-axis at (0, 6) and the x-axis at (2, 0) and the area to the left is shaded. A solid line with a positive slope passes through the origin (0, 0) and the area to the right is shaded. Shaded sections that overlap are a little in quadrant 1, half of quadrant 3, and a little less than half of quadrant 4.

 

D)

A coordinate plane shows two dashed lines that intersect, with 3 shaded areas. A dashed line with a negative slope passes the y-axis at (0, 6) and the x-axis at (2, 0) and the area to the left is shaded. A dashed line with a positive slope passes through the origin (0, 0) and the area to the right is shaded. Shaded sections that overlap are a little in quadrant 1, half of quadrant 3, and a little less than half of quadrant 4.

 

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Graphing a system of inequalities can help you solve real-life situations.

Example

Problem

Cathy is selling ice cream cones at a school fundraiser. She is selling two sizes: small (which has `1` scoop) and large (which has `2` scoops). She knows that she can get a maximum of `70` scoops of ice cream out of her supply. She charges `$3` for a small cone and `$5` for a large cone.

 

Cathy wants to earn at least `$120` to give back to the school. Write and graph a system of inequalities that models this situation.

 

`s =` small cone

`l =` large cone

First, identify the variables. There are two variables: the number of small cones and the number of large cones.

 

`s + 2l<=70`

 

 

Write the first equation: the maximum number of scoops she can give out. The scoops she has available `(70)` must be greater than or equal to the number of scoops for the small cones `(s)` and the large cones `(2l)` she sells.

 

`3s + 5l>=120`

 

Write the second equation: the amount of money she raises. She wants the total amount of money earned from small cones `(3s)` and large cones `(5l)` to be at least `$120`.

 

`s + 2l<=70`

`3s + 5l>=120`

Write the system.

 

Now graph the system. The variables `x` and `y` have been replaced by `s` and `l`; graph `s` along the `x`-axis, and `l` along the `y`-axis.

 

First, graph the region `s + 2l<=70`. Graph the boundary line and then test individual points to see which region to shade. The graph is shown below.

A graph shows Quadrant 1 of a coordinate plane with Number of Small Cones (s) on the x-axis and Number of Large Cones (l) on the y-axis. A solid line with a negative slope crosses the y-axis at (0, 35) and the x-axis at (70, 0). The area below and to the left of the line is shaded red. The inequality S plus 2 L is less than or equal to 70 is shown in the shaded area.

 

Now, graph the region `3s + 5l>=120`. Graph the boundary line and then test individual points to see which region to shade. The graph is shown below.

A graph shows Quadrant 1 of a coordinate plane with Number of Small Cones (s) on the x-axis and Number of Large Cones (l) on the y-axis. A solid line with a negative slope crosses the y-axis at (0, 24) and the x-axis at (40, 0). The area above and to the right of the line is shaded blue. The inequality 3 S plus 5 L is greater than or equal to 120 is shown in the shaded area.

 

Graphing the regions together, you find the following:

A graph shows Quadrant 1 of a coordinate plane with Number of Small Cones (s) on the x-axis and Number of Large Cones (l) on the y-axis. A solid line with a negative slope crosses the y-axis at (0, 35) and the x-axis at (70, 0). The area below and to the left of the line is shaded. Another solid line with a negative slope crosses the y-axis at (0, 24) and the x-axis at (40, 0). The area above and to the right of the line is shaded. The shaded areas between the lines overlap and shows the inequalities, 3 S plus 5 L is greater than or equal to 120 and S plus 2 L is less than or equal to 70.

And represented just as the overlapping region, you have:

 

A graph shows Quadrant 1 of a coordinate plane with Number of Small Cones (s) on the x-axis and Number of Large Cones (l) on the y-axis. A solid line with a negative slope crosses the y-axis at (0, 35) and the x-axis at (70, 0). Another solid line with a negative slope crosses the y-axis at (0, 24) and the x-axis at (40, 0). The area between the lines is shaded purple. The inequalities 3 S plus 5 L is greater than or equal to 120 and S plus 2 L is less than or equal to 70 are shown in the shaded area.

Answer

The region in purple is the solution. As long as the combination of small cones and large cones that Cathy sells can be mapped in the purple region, she will have earned at least `$120` and not used more than `70` scoops of ice cream.

Summary

Systems of inequalities can be graphed on a coordinate plane. The solution set for a system of inequalities is not a single point, but rather an entire region defined by the overlapping areas of each individual inequality in the system. Every point within this region is a solution to both inequalities and thus a solution for the whole system.