4.7 Graphs of linear inequalities

<< Chapter < Page

Elementary algebra

Page 1 / 10

Chapter >> Page >

By the end of this section, you will be able to:

Verify solutions to an inequality in two variables
Recognize the relation between the solutions of an inequality and its graph
Graph linear inequalities

Before you get started, take this readiness quiz.

Solve: $4 x + 3 > 23$ .
If you missed this problem, review [link] .
Translate from algebra to English: $x < 5$ .
If you missed this problem, review [link] .
Evaluate $3 x - 2 y$ when $x = 1$ , $y = −2$ .
If you missed this problem, review [link] .

Verify solutions to an inequality in two variables

We have learned how to solve inequalities in one variable. Now, we will look at inequalities in two variables. Inequalities in two variables have many applications. If you ran a business, for example, you would want your revenue to be greater than your costs—so that your business would make a profit.

Linear inequality

A linear inequality is an inequality that can be written in one of the following forms:

\begin{matrix} A x + B y > C & A x + B y \geq C & A x + B y < C & A x + B y \leq C \end{matrix}

where $A and B$ are not both zero.

Do you remember that an inequality with one variable had many solutions? The solution to the inequality $x > 3$ is any number greater than 3. We showed this on the number line by shading in the number line to the right of 3, and putting an open parenthesis at 3. See [link] .

The figure shows a number line extending from negative 5 to 5. A parenthesis is shown at positive 3 and an arrow extends form positive 3 to positive infinity.

Similarly, inequalities in two variables have many solutions. Any ordered pair $(x, y)$ that makes the inequality true when we substitute in the values is a solution of the inequality.

Solution of a linear inequality

An ordered pair $(x, y)$ is a solution of a linear inequality if the inequality is true when we substitute the values of x and y .

Determine whether each ordered pair is a solution to the inequality $y > x + 4$ :

ⓐ $(0, 0)$ ⓑ $(1, 6)$ ⓒ $(2, 6)$ ⓓ $(−5, −15)$ ⓔ $(−8, 12)$

ⓐ

$(0, 0)$

Simplify.
So, $(0, 0)$ is not a solution to $y > x + 4$ .
ⓑ

$(1, 6)$

Simplify.
So, $(1, 6)$ is a solution to $y > x + 4$ .
ⓒ

$(2, 6)$

Simplify.
So, $(2, 6)$ is not a solution to $y > x + 4$ .
ⓓ

$(−5, −15)$

Simplify.
So, $(−5, −15)$ is not a solution to $y > x + 4$ .
ⓔ

$(−8, 12)$

Simplify.
So, $(−8, 12)$ is a solution to $y > x + 4$ .

Got questions? Get instant answers now!

Determine whether each ordered pair is a solution to the inequality $y > x - 3$ :

ⓐ $(0, 0)$ ⓑ $(4, 9)$ ⓒ $(−2, 1)$ ⓓ $(−5, −3)$ ⓔ $(5, 1)$

ⓐ yes ⓑ yes ⓒ yes ⓓ yes ⓔ no

Got questions? Get instant answers now!

Determine whether each ordered pair is a solution to the inequality $y < x + 1$ :

ⓐ $(0, 0)$ ⓑ $(8, 6)$ ⓒ $(−2, −1)$ ⓓ $(3, 4)$ ⓔ $(−1, −4)$

ⓐ yes ⓑ yes ⓒ no ⓓ no ⓔ yes

Got questions? Get instant answers now!

Recognize the relation between the solutions of an inequality and its graph

Now, we will look at how the solutions of an inequality relate to its graph.

Let’s think about the number line in [link] again. The point $x = 3$ separated that number line into two parts. On one side of 3 are all the numbers less than 3. On the other side of 3 all the numbers are greater than 3. See [link] .

The solution to $x > 3$ is the shaded part of the number line to the right of $x = 3$ .

Similarly, the line $y = x + 4$ separates the plane into two regions. On one side of the line are points with $y < x + 4$ . On the other side of the line are the points with $y > x + 4$ . We call the line $y = x + 4$ a boundary line.

Boundary line

The line with equation $A x + B y = C$ is the boundary line that separates the region where $A x + B y > C$ from the region where $A x + B y < C$ .

For an inequality in one variable, the endpoint is shown with a parenthesis or a bracket depending on whether or not $a$ is included in the solution:

Questions & Answers

what does preconceived mean

sammie Reply

physiological Psychology

Nwosu Reply

How can I develope my cognitive domain

Amanyire Reply

why is communication effective

Dakolo Reply

Communication is effective because it allows individuals to share ideas, thoughts, and information with others.

effective communication can lead to improved outcomes in various settings, including personal relationships, business environments, and educational settings. By communicating effectively, individuals can negotiate effectively, solve problems collaboratively, and work towards common goals.

it starts up serve and return practice/assessments.it helps find voice talking therapy also assessments through relaxed conversation.

miss

Every time someone flushes a toilet in the apartment building, the person begins to jumb back automatically after hearing the flush, before the water temperature changes. Identify the types of learning, if it is classical conditioning identify the NS, UCS, CS and CR. If it is operant conditioning, identify the type of consequence positive reinforcement, negative reinforcement or punishment

Wekolamo Reply

please i need answer

Wekolamo

because it helps many people around the world to understand how to interact with other people and understand them well, for example at work (job).

Manix Reply

Agreed 👍 There are many parts of our brains and behaviors, we really need to get to know. Blessings for everyone and happy Sunday!

ARC

A child is a member of community not society elucidate ?

JESSY Reply

Isn't practices worldwide, be it psychology, be it science. isn't much just a false belief of control over something the mind cannot truly comprehend?

Simon Reply

compare and contrast skinner's perspective on personality development on freud

namakula Reply

Skinner skipped the whole unconscious phenomenon and rather emphasized on classical conditioning

war

explain how nature and nurture affect the development and later the productivity of an individual.

Amesalu Reply

nature is an hereditary factor while nurture is an environmental factor which constitute an individual personality. so if an individual's parent has a deviant behavior and was also brought up in an deviant environment, observation of the behavior and the inborn trait we make the individual deviant.

Samuel

I am taking this course because I am hoping that I could somehow learn more about my chosen field of interest and due to the fact that being a PsyD really ignites my passion as an individual the more I hope to learn about developing and literally explore the complexity of my critical thinking skills

Zyryn Reply

good👍

Jonathan

and having a good philosophy of the world is like a sandwich and a peanut butter 👍

Jonathan

generally amnesi how long yrs memory loss

Kelu Reply

interpersonal relationships

Abdulfatai Reply

What would be the best educational aid(s) for gifted kids/savants?

Heidi Reply

treat them normal, if they want help then give them. that will make everyone happy

Saurabh

What are the treatment for autism?

Magret Reply

hello. autism is a umbrella term. autistic kids have different disorder overlapping. for example. a kid may show symptoms of ADHD and also learning disabilities. before treatment please make sure the kid doesn't have physical disabilities like hearing..vision..speech problem. sometimes these

Jharna

continue.. sometimes due to these physical problems..the diagnosis may be misdiagnosed. treatment for autism. well it depends on the severity. since autistic kids have problems in communicating and adopting to the environment.. it's best to expose the child in situations where the child

Jharna

child interact with other kids under doc supervision. play therapy. speech therapy. Engaging in different activities that activate most parts of the brain.. like drawing..painting. matching color board game. string and beads game. the more you interact with the child the more effective

Jharna

results you'll get.. please consult a therapist to know what suits best on your child. and last as a parent. I know sometimes it's overwhelming to guide a special kid. but trust the process and be strong and patient as a parent.

Jharna

Got questions? Join the online conversation and get instant answers!

Jobilize.com Reply

<< Chapter < Page Page > Chapter >>

Practice Key Terms 3

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

100% Free Mobile Applications
Receive real-time job alerts and never miss the right job again

Source: OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2

Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Elementary algebra' conversation and receive update notifications?

Ask

©flickr:	Anatomy Physiology By Jemekia Weeden Start Quiz
	17 Biology 17 Biotechnology and Genomics MCQ By OpenStax Start Quiz
	Anthropology Language Culture By Richley Crapo Start Assignment
	Western Political Thought MCQ By Saylor Foundation Start Quiz
	18 AP 18 Cardiovascular System Blood MCQ By OpenStax Start Quiz
	1 Neuroanatomy 01 The Cranial Nerves By Stephen Voron Start Quiz
	Grade 10 Module 2.1 IT Quiz (Part 1) By Christine Zeelie Start Quiz
	College physics for ap® courses By OpenStax Read Online Course
	Are you part of the R5 Family? By Anonymous User Start Quiz
	1 Lec:1 Descriptive Epidemiology By Janet Forrester Start Quiz

$(0, 0)$

Simplify.	So, $(0, 0)$ is not a solution to $y > x + 4$ .

$(1, 6)$

Simplify.	So, $(1, 6)$ is a solution to $y > x + 4$ .

$(2, 6)$

Simplify.	So, $(2, 6)$ is not a solution to $y > x + 4$ .

$(−5, −15)$

Simplify.	So, $(−5, −15)$ is not a solution to $y > x + 4$ .

$(−8, 12)$

Simplify.	So, $(−8, 12)$ is a solution to $y > x + 4$ .