Math resources Algebra Inequalities

Graphing inequalities

# Graphing inequalities

Here you will learn about graphing inequalities, including what they look like on a graph, horizontal lines, vertical lines, systems of inequalities and shading regions.

Students will first learn about graphing inequalities as a part of expressions and equations in grade 6 and will expand on that knowledge into high school math.

## What are graphing inequalities?

Graphing inequalities allows you to visualize the regions that satisfy one or more inequalities.

These inequalities are often linear and can be expressed using straight line graphs. You may have to use graphs already provided to find solutions to the inequalities or you may need to draw lines and indicate a region that satisfies the system of inequalities.

To do this, use the linear equations to plot straight line graphs using a boundary line, which is either a solid line or a dashed line.

A solid line means the line is included.

A dashed/dotted line means the line is not included.

This is similar to using the solid (or closed) circle and open circles when displaying inequalities on a number line.

It is important to indicate the region required using the method requested in the question.

The question may ask you to shade a region required, it may ask you to indicate the region with a letter or it may ask you to indicate integer coordinates that satisfy a system of inequalities with crosses.

To get the correct region, think about what coordinates will satisfy the inequality.

For horizontal inequality lines in the form y<a or y>a, you need to think about what the y -coordinate could be.

For lines that are not vertical or horizontal you can use the same thinking to find the correct region, or substitute a coordinate, such as the origin (0, 0) into the inequality to test the validity.

### System of inequalities

For a system of inequalities you need to draw the regions that satisfy all of the inequalities stated.

For example:

## Common Core State Standards

How does this relate to 6 th grade math and high school math?

• Grade 6: Expressions and Equations (6.EE.B.8)
Write an inequality of the form x>c or x<c to represent a constraint or condition in a real-world or mathematical problem.

Recognize that inequalities of the form x>c or x<c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.

• High School: Algebra (HS.A.REI.D.12)
Graph the solutions to a linear inequality in two variables as a half plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

## How to graph inequalities

In order to graph inequalities:

1. Find a set of coordinates that satisfy a line given by the inequality.
2. Join the points using a dashed line or a solid line.
3. Indicate the points that satisfy the inequality.

## Graphing inequalities examples

### Example 1: shading a region for a single inequality

Shade the region that satisfies the inequality x>- \, 4.

1. Find a set of coordinates that satisfy a line given by the inequality.

You need points on the line x>- \, 4.

These are ordered pairs such as, (- \, 4,- \, 3), (- \, 4,0), (- \, 4,2), …

2Join the points using a dashed line or a solid line.

x>- \, 4 will require a dashed line as the inequality is only greater than - \, 4, not equal to.

3Indicate the points that satisfy the inequality.

You want the values of x that are greater than - \, 4, so shade the right hand side of the line.

### Example 2: shade a region between two inequalities

Shade the region that satisfies the inequality - \, 3\leq{y}<1.

Find a set of coordinates that satisfy a line given by the inequality.

Join the points using a dashed line or a solid line.

Indicate the points that satisfy the inequality.

### Example 3: shade the region for an inequality with a line in the form y = mx + c

Shade the region that satisfies y\geq{2x}-1.

Find a set of coordinates that satisfy a line given by the inequality.

Join the points using a dashed line or a solid line.

Indicate the points that satisfy the inequality.

### Example 4: indicate a region for an inequality with a line in standard form ax + by = c

Indicate the region that satisfies the inequality 4x+3y<24 with an R.

Find a set of coordinates that satisfy a line given by the inequality.

Join the points using a dashed line or a solid line.

Indicate the points that satisfy the inequality.

### Example 5: indicating a region that satisfies a system of inequalities

Mark with a cross (x) the integer coordinates that satisfy

x+y>5, x<2 and y\leq{7}.

Find a set of coordinates that satisfy a line given by the inequality.

Join the points using a dashed line or a solid line.

Indicate the points that satisfy the inequality.

### Teaching tips for graphing inequalities

• Use visual aids, such as number lines and graphs, to allow students to visually see the inequalities. The use of color coding when first introducing graphing inequalities allows for better visual distinction with the boundary lines and shaded areas.

• Incorporate appropriate technology when working with students. There are numerous graphing calculators and software available online that can be another tool for students. Interactive whiteboards are an amazing way to engage students while graphing inequalities.

• Providing students with timely feedback and support is important to not create misconceptions. Technology tools can be used for students to check their work in real time.

### Easy mistakes to make

• Indicating the wrong side of the line with equations involving \textbf{y}
It is common to indicate the wrong side of the line that satisfies an inequality involving the variable y. The sight of a positive y> means it will be above the line, a positive y< means it will be below the line.

• Using the wrong type of line
When drawing lines it is important to use a dashed line for inequalities in using the symbol < or >. There may be questions using these symbols with solid lines already drawn – this sort of question will usually want you to indicate integer coordinates that satisfy the inequality.

• Indicating that points on a dashed line will satisfy the inequality
When dealing with inequalities that are strictly less than or greater than ( indicated by the symbol < or >), the points on the line are not included.

### Practice graphing inequalities questions

1. The diagram shows a shaded region satisfying an inequality.

Select the correct inequality.

y\leq{3}

x>3

y<3

y>3

A horizontal dashed line means that the answer will not include x. The region that is shaded is below the line, meaning that the inequality is representing all of the values of y less than 3.

2. The diagram shows a shaded region satisfying an inequality.

Select the correct inequality.

x\leq{- \, 4}

x>- \, 4

y>- \, 4

x\geq{- \, 4}

A vertical solid line means that the answer will include x. The region that is shaded is to the right side of the line, meaning that the inequality is representing all of the values greater than – \,4.

3. The diagram shows a shaded region satisfying an inequality.

Select the correct inequality.

1<y<3

1<y\leq{3}

1\leq{y}\leq{3}

1\leq{y}<3

The inequality contains two horizontal dashed lines, meaning the values are not included. The region between 1 and 3 is shaded, meaning the value of y must be between the two values.

4. The diagram shows a shaded region satisfying an inequality.

Select the correct inequality.

y<x+3

y>x+3

y\leq{x}+3

y\geq{x}+3

The solid line means that the answer will include the value. The region that is shaded is below the line, meaning that the inequality is representing all of the values less than only.

5. Which diagram indicates the region satisfied by the inequalities:
y\geq{x}, x\leq{4} and y\geq{1} ?

The region must be above the line y=x, the left of the line x=4 and above the line y=1.

6. Which diagram indicates the region satisfied by the inequalities:
y\geq{2}, 2x+y\leq{4} and x\geq{- \, 1}?

The region must be below the line 2x+y=4, above the line y=2 and to the right of the line x=- \, 1.

## Graphing inequalities FAQs

What is an inequality?

An inequality is a mathematical statement that compares two expressions using inequality signs and symbols (‘<’, ‘>’, “\leq” \, “\geq”).

What are systems of inequalities?

Similar to a system of equations, systems of inequalities is a set of two or more inequalities with the same variables. The solution to the system is the region where the shaded area of all the inequalities graphed intersect, or overlap. A system of equations uses an equal sign and a system of inequalities uses inequality signs.

How do you graph a linear inequality?

Start with a linear equality written in slope-intercept form (y=mx+b). Next you will graph the boundary line, using either a solid line for “\leq” \, “\geq” or dashed line for ‘<’, ‘>’.

Then you will shade the appropriate region, if inequality y is less than or less than and equal to sign, shade below the line and if the inequality is greater than or greater than and equal to, shade above the line.

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