Topics
2. Equations & Inequalities in Two Variables
topic badge
United States of AmericaTennessee
Integrated Math 1

2.04 Linear inequalities in two variables

Lesson
Practice
Lesson

Concept summary

If a linear inequality involves two variables, we can represent it as a region on a coordinate plane rather than an interval on a number line.

Linear inequality in two variables

An inequality whose solution is a set of ordered pairs represented by a region of the coordinate plane on one side of a boundary line.

Example:

2y > 4 - 3x

Boundary line

A line which divides the coordinate plane into two regions. A boundary line of a linear inequality is solid if it is included in the solution set, and dashed if it is not.

x
y

Depending on the inequality sign, the boundary line will be solid or dashed, and region shaded will be above or below the boundary line.

-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
y>x
-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
y \geq x
-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
y<x
-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
y \leq x

Worked examples

Example 1

Write the inequality that describes the region shaded on the given coordinate plane.

-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y

Approach

First, we want to determine the equation of the boundary line. Then we need to determine what inequality sign to use.

Solution

-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y

To find the equation of the boundary line in the form y=mx+b:

  • The line crosses the y-axis at (0,-3)
  • The rise is 3
  • The run is 1

So, b=-3 and m=3.

Now we have y=3x-3 as the boundary line.

-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y

To determine the inequality:

  • The boundary line is solid, so it is \leq or \geq.
  • Any point in the shaded region would be above the boundary line.

This means that for every x-value, the y-value for any point in the region is greater than the y-value of the point on the line.

So our linear inequality is y \geq 3x-3.

Reflection

It is a good idea to check our answer by substituting in a point in the region to make sure it satisfies the inequality. For example, the origin, \left(0,0\right), is in the region, so should satisfy the inequality.

\displaystyle y\displaystyle \geq\displaystyle 3x-3
\displaystyle 0\displaystyle \geq\displaystyle 3\left(0\right)-3
\displaystyle 0\displaystyle \geq\displaystyle -3

It does satisfy the inequality, so we have selected the correct inequality sign.

Example 2

A pick-up truck has a maximum weight capacity of 3000 pounds. Each box of oranges weighs 8 pounds and each box of grapefruits weighs 12 pounds.

Let x represents the number of boxes of oranges in the truck. Let y represents the number of boxes of grapefruit in the truck.

a

Write an inequality to represent the number of boxes of oranges and grapefruit that can be in the truck.

Solution

The weight of all the orange boxes is the product of the weight of one box, 8, and the number of boxes, x.

Weight of orange boxes: 8x

The weight of all the grapefruit boxes is the product of the weight of one box, 12, and the number of boxes, y.

Weight of grapefruit boxes: 12y

The total weight is the sum of the weights of orange boxes and grapefruit boxes.

Total weight: 8x+12y

The truck can carry at most 3000 pounds, so we get our inequality:

8x+12y \leq 3000

b

Create a graph of the region containing the points corresponding to all the different numbers of orange and grapefruit boxes that can be loaded into the truck.

Approach

We will graph the region representing 8x+12y \leq 3000.

We need to identify our boundary line, then graph it. Then, we need to decide which side of the boundary line to shade.

Solution

The boundary line is 8x+12y= 3000.

This is a line in standard form, so we can graph it by finding the intercepts.

Find the x-intercept by setting y=0 and solving:

\displaystyle 8x+12y\displaystyle =\displaystyle 3000Stating the given equation
\displaystyle 8x-12\left(0\right)\displaystyle =\displaystyle 3000Setting y=0
\displaystyle 8x\displaystyle =\displaystyle 3000Simplifying
\displaystyle x\displaystyle =\displaystyle 375Dividing both sides by 8

Find the y-intercept by setting x=0 and solving:

\displaystyle 8x+12y\displaystyle =\displaystyle 3000Stating the given equation
\displaystyle 8\left(0\right)+12y\displaystyle =\displaystyle 3000Setting x=0
\displaystyle 12y\displaystyle =\displaystyle 3000Simplifying
\displaystyle y\displaystyle =\displaystyle 250Dividing both sides by 12

The boundary line will be solid because we have \leq as the sign.

We will shade below the line as the point \left(0,0\right) satisfies the inequality.

25
50
75
100
125
150
175
200
225
250
275
300
325
350
375
400
x
25
50
75
100
125
150
175
200
225
250
275
300
y

Reflection

The actual possible solutions are the positive integer coordinates since we are working with whole fruits and have both x\geq 0 and y \geq 0. This region shows where all those possible solutions can be.

Outcomes

M1.N.Q.A.1

Use units as a way to understand real-world problems.*

M1.N.Q.A.1.A

Choose and interpret the scale and the origin in graphs and data displays.*

M1.A.CED.A.3

Create individual and systems of equations and/or inequalities to represent constraints in a contextual situation, and interpret solutions as viable or non-viable.*

M1.MP1

Make sense of problems and persevere in solving them.

M1.MP2

Reason abstractly and quantitatively.

M1.MP3

Construct viable arguments and critique the reasoning of others.

M1.MP4

Model with mathematics.

M1.MP5

Use appropriate tools strategically.

M1.MP6

Attend to precision.

M1.MP7

Look for and make use of structure.

M1.MP8

Look for and express regularity in repeated reasoning.

What is Mathspace

About Mathspace