2.7 Solve linear inequalities (Page 2/8)

Page 2 / 8

The inequality $x \leq 1$ means all numbers less than or equal to 1. There is no lower end to those numbers. We write $x \leq 1$ in interval notation as $(− \infty, 1]$ . The symbol $− \infty$ is read as ‘negative infinity’. [link] shows both the number line and interval notation.

This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than or equal to 1 is graphed on the number line, with an open bracket at x equals 1, and a red line extending to the left of the bracket. The inequality is also written in interval notation as parenthesis, negative infinity comma 1, bracket. — The inequality $x \leq 1$ is graphed on this number line and written in interval notation.

Inequalities, number lines, and interval notation

This figure show four number lines, all without tick marks. The inequality x is greater than a is graphed on the first number line, with an open parenthesis at x equals a, and a red line extending to the right of the parenthesis. The inequality is also written in interval notation as parenthesis, a comma infinity, parenthesis. The inequality x is greater than or equal to a is graphed on the second number line, with an open bracket at x equals a, and a red line extending to the right of the bracket. The inequality is also written in interval notation as bracket, a comma infinity, parenthesis. The inequality x is less than a is graphed on the third number line, with an open parenthesis at x equals a, and a red line extending to the left of the parenthesis. The inequality is also written in interval notation as parenthesis, negative infinity comma a, parenthesis. The inequality x is less than or equal to a is graphed on the last number line, with an open bracket at x equals a, and a red line extending to the left of the bracket. The inequality is also written in interval notation as parenthesis, negative infinity comma a, bracket.

Did you notice how the parenthesis or bracket in the interval notation matches the symbol at the endpoint of the arrow? These relationships are shown in [link] .

This figure shows the same four number lines as above, with the same interval notation labels. Below the interval notation for each number line, there is text indicating how the notation on the number lines is similar to the interval notation. The first number line is a graph of x is greater than a, and the interval notation is parenthesis, a comma infinity, parenthesis. The text below reads: “Both have a left parenthesis.” The second number line is a graph of x is greater than or equal to a, and the interval notation is bracket, a comma infinity, parenthesis. The text below reads: “Both have a left bracket.” The third number line is a graph of x is less than a, and the interval notation is parenthesis, negative infinity comma a, parenthesis. The text below reads: “Both have a right parenthesis.” The last number line is a graph of x is less than or equal to a, and the interval notation is parenthesis, negative infinity comma a, bracket. The text below reads: “Both have a right bracket.” — The notation for inequalities on a number line and in interval notation use similar symbols to express the endpoints of intervals.

Graph on the number line and write in interval notation.

ⓐ $x \geq −3$ ⓑ $x < 2.5$ ⓒ $x \leq - \frac{3}{5}$

Solution

ⓐ

Shade to the right of $−3$ , and put a bracket at $−3$ .

Write in interval notation.
ⓑ

Shade to the left of $2.5$ , and put a parenthesis at $2.5$ .

Write in interval notation.
ⓒ

Shade to the left of $- \frac{3}{5}$ , and put a bracket at $- \frac{3}{5}$ .

Write in interval notation.

Got questions? Get instant answers now!

Graph on the number line and write in interval notation:

ⓐ $x > 2$ ⓑ $x \leq - 1.5$ ⓒ $x \geq \frac{3}{4}$

Got questions? Get instant answers now!

Graph on the number line and write in interval notation:

ⓐ $x \leq - 4$ ⓑ $x \geq 0.5$ ⓒ $x < - \frac{2}{3}$

Got questions? Get instant answers now!

Solve inequalities using the subtraction and addition properties of inequality

The Subtraction and Addition Properties of Equality state that if two quantities are equal, when we add or subtract the same amount from both quantities, the results will be equal.

Properties of equality

\begin{matrix} Subtraction Property of Equality & Addition Property of Equality \\ For any numbers a, b, and c, & For any numbers a, b, and c, \\ \begin{matrix} if & a & = & b, \\ then & a - c & = & b - c . \end{matrix} & \begin{matrix} if & a & = & b, \\ then & a + c & = & b + c . \end{matrix} \end{matrix}

Similar properties hold true for inequalities.

For example, we know that −4 is less than 2.
If we subtract 5 from both quantities, is the left side still less than the right side?
We get −9 on the left and −3 on the right.
And we know −9 is less than −3.
	The inequality sign stayed the same.

Similarly we could show that the inequality also stays the same for addition.

This leads us to the Subtraction and Addition Properties of Inequality.

Properties of inequality

\begin{matrix} Subtraction Property of Inequality & Addition Property of Inequality \\ For any numbers a, b, and c, & For any numbers a, b, and c, \\ \begin{matrix} if & a & < & b \\ then & a - c & < & b - c . \\ if & a & > & b \\ then & a - c & > & b - c . \end{matrix} & \begin{matrix} if & a & < & b \\ then & a + c & < & b + c . \\ if & a & > & b \\ then & a + c & > & b + c . \end{matrix} \end{matrix}

We use these properties to solve inequalities, taking the same steps we used to solve equations. Solving the inequality $x + 5 > 9$ , the steps would look like this:

\begin{matrix} x + 5 & > & 9 \\ Subtract 5 from both sides to isolate x . & x + 5 - 5 & > & 9 - 5 \\ Simplify. & x & > & 4 \end{matrix}

Any number greater than 4 is a solution to this inequality.

Solve the inequality $n - \frac{1}{2} \leq \frac{5}{8}$ , graph the solution on the number line, and write the solution in interval notation.

Solution


Add $\frac{1}{2}$ to both sides of the inequality.
Simplify.
Graph the solution on the number line.
Write the solution in interval notation.

Got questions? Get instant answers now!

Solve the inequality, graph the solution on the number line, and write the solution in interval notation.

$p - \frac{3}{4} \geq \frac{1}{6}$

This figure shows the inequality p is greater than or equal to 11/12. Below this inequality is the inequality graphed on a number line ranging from 0 to 4, with tick marks at each integer. There is a bracket at p equals 11/12, and a dark line extends to the right from 11/12. Below the number line is the solution written in interval notation: bracket, 11/12 comma infinity, parenthesis.

Got questions? Get instant answers now!

Solve the inequality, graph the solution on the number line, and write the solution in interval notation.

$r - \frac{1}{3} \leq \frac{7}{12}$

This figure shows the inequality r is less than or equal to 11/12. Below this inequality is the inequality graphed on a number line ranging from 0 to 4, with tick marks at each integer. There is a bracket at r equals 11/12, and a dark line extends to the left from 11/12. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma 11/12, bracket.

Got questions? Get instant answers now!

Solve inequalities using the division and multiplication properties of inequality

The Division and Multiplication Properties of Equality state that if two quantities are equal, when we divide or multiply both quantities by the same amount, the results will also be equal (provided we don’t divide by 0).

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 >>

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

	10 Clin Sci I - GI Twedt quiz By Brooke Delaney Start Quiz
	6 BOD Respiratory Exam By Brooke Delaney Start Exam
	Journalism Midterm By Sheila Lopez Start Exam
	Cultural Anthropology Assignment 2 By Richley Crapo Start Assignment
	8 AP 08 Appendicular Skeleton Essay By OpenStax Start Flashcards
	45 Biology 45 Population and Community Ecology MCQ By OpenStax Start Quiz
	2 Understanding Societies 2 By Jessica Collett Start Exam
	U.s. history By OpenStax Read Online Course
	2 AP 02 Chemical Level of Organization Essay By OpenStax Start Flashcards
	Computer System Engineering By Robert Morris Start Exam


Shade to the right of $−3$ , and put a bracket at $−3$ .
Write in interval notation.


Shade to the left of $2.5$ , and put a parenthesis at $2.5$ .
Write in interval notation.


Shade to the left of $- \frac{3}{5}$ , and put a bracket at $- \frac{3}{5}$ .
Write in interval notation.