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Inequalities

For in GOD we live, and move, and have our being. - Acts 17:28

The Joy of a Teacher is the Success of his Students. - Samuel Dominic Chukwuemeka

Welcome to Our Site

Samdom For Peace

I greet you this day,
First: review the prerequisite topics. Second: read the notes. Third: view the videos. Fourth: solve the questions/solved examples. Fifth: check your solutions with my thoroughly-explained solutions. Sixth: check your answers with the calculators as applicable.
The Wolfram Alpha widgets (many thanks to the developers) was used for the inequalities calculators.
Comments, ideas, areas of improvement, questions, and constructive criticisms are welcome. You may contact me.
If you are my student, please do not contact me here. Contact me via the school's system.
Thank you for visiting!!!

Samuel Dominic Chukwuemeka (Samdom For Peace) B.Eng., A.A.T, M.Ed., M.S

Objectives

Objectives

Students will:

(1.) Define inequalities.

(2.) Discuss inequality in the world.

(3.) Discuss inequality in mathematics.

(4.) Solve linear inequalities.

(5.) Check the solution(s) of linear inequalities.

(6.) Graph the solution(s) of linear inequalities.

(7.) Solve quadratic inequalities.

(8.) Check the solution(s) of quadratic inequalities.

(9.) Graph the solution(s) of quadratic inequalities.

(10.) Solve polynomial inequalities.

(11.) Check the solution(s) of polynomial inequalities.

(12.) Graph the solution(s) of polynomial inequalities.

(13.) Solve absolute value inequalities.

(14.) Check the solution(s) of absolute value inequalities.

(15.) Graph the solution(s) of absolute value inequalities.

(16.) Solve rational inequalities.

(17.) Check the solution(s) of rational inequalities.

(18.) Graph the solution(s) of rational inequalities.


Skills Measured/Acquired

(1.) Use of prior knowledge

(2.) Critical Thinking

(3.) Interdisciplinary connections/applications

(4.) Technology

(5.) Active participation through direct questioning

(6.) Research

Vocabulary Words

Please note:
added to
subtracted from
multiplied by
divided by

Check for prior knowledge. Ask students about these terms.

Bring it to English: equal, unequal, equality, inequality, more, less, greater, smaller, lesser, most, least, at most, at least, exactly, just, not


Bring it to Mathematics: inequality, greater than, less than, greater than or equal to, less than or equal to, at least, at most, exactly, equal, equals, equal to, not equal to


(1.) At least $5$ means $\ge 5$
It means that the minimum should be $5$

(2.) At most $5$ means $\le 5$
It means that the maximum should be $5$

(3.) Just $5$ means $= 5$
It means exactly $5$ or equal to $5$

(4.) More than $5$ means $\gt 5$

(5.) Less than $5$ means $\lt 5$

(6.) No more than $5$ means $\le 5$
It should not be more than $5$
It means $5$ or less

(7.) No less than $5$ means $\ge 5$
It should not be less than $5$
It means $5$ or more

Introduction

Thoughtful Scenario!

Teacher: What is the root of all evil?
Student: Inequality
Teacher: Hmmmm...no...
Student: What is it?
Teacher: The love of money.
Student: How do you know?
Teacher: $1$ Timothy $6:10$

Teacher: What is the root of most social problems?
Students: Inequality.
Teacher: Correct!
Inequality does not only create problems in the real world.
It creates problems in Mathematics too!
Student: How does it create problem in Mathematics?
Teacher: Good question.
Remember when you check your work after solving equations...
Student: To make sure the $LHS = RHS$?...
Teacher. Right....
And if the $LHS \ne RHS$,
Student: The solution is incorrect. You go back and see where you made a mistake, fix it, and check again.
Teacher: Right.
In Inequality, we shall still check our work.
However, it is possible for the incorrect solution to give a right check.
It is possible for the solution(s) to satisfy the "Check" even though it is incorrect.
Student: How are we certain to know the correct solutions then?
Teacher: That is why equations are great. Equality is great.
When you treat everyone equally, the world is safe.
When there is inequality anywhere, it is a threat to equality everywhere.
Student: The last statement seems familiar to a quote by MLK.
Teacher: What is the quote?
Student: "Injustice anywhere is a threat to justice everywhere" - Martin Luther King Jr.
Teacher: Yes! The same applies to inequality.
So, there is really no way to ensure certainty of the correctness of your solution and avoid incorrect solutions that also checks because of "some intervals" that satisfy that inequality.

Teacher: So, if I am not equal to you, that means ....
Student: You are "greater than" me or "less than" me
Teacher: The "less than" sign looks like a bent "l"...l as in lion
The other sign is the "greater than" sign. It looks like a horizontal reflection of it.
Student: What is a horizontal reflection?
Teacher: We shall do it later in this course...when we do Algebra Transformations.
It is "Same $y$-value for opposite $x$-values".

Student: English Language is kinda funny...
Teacher: How is that?
Student: The opposite of equal is unequal
One would think the opposite of equality would be "un"equality rather than inequality
Teacher: Hmmm...interesting...


Three Notable Notes When Solving Inequalities

There are three major things to note when solving Inequalities:
Let us first illustrate why equality (equation) is much better than inequality.

Equation
$3 = 3$
$-3 = -3$
Whether you multiply or divide both sides of an equation by $-1$, the equation is still the same.

Inequality
$3 \lt 4$
Multiply both sides by $-1$
$-1 * 3 \gt -1 * 4$
$-3 \gt -4$
Teacher: Did you notice what happened?
Student: The inequality sign is reversed.
Teacher: That's right.

(1.) Whenever you multiply or divide both sides of an inequality by a negative number, the inequality is reversed.


Equation
$2 + 3 = 5$
$5 = 2 + 3$
The $LHS$ is equal to the $RHS$
The $RHS$ is equal to the $LHS$
You can swap: make the $LHS$ to be the $RHS$ and the $RHS$ to be the $LHS$ and there is no problem.

Inequality
$(2 + 3) \lt 6$
$6 \gt (2 + 3)$
Teacher: Did you notice what happened?
Student: The inequality sign is reversed.
Teacher: That's right.

(2.) Whenever you swap (make the $LHS$ to be the $RHS$, and the $RHS$ to be the $LHS$); the inequality is reversed.


Equation
$3 = 3$

$\dfrac{1}{3} = \dfrac{1}{3}$

Whether you take the reciprocal of both sides of an equation, the equation remains the same.

Inequality
$3 \lt 4$
Take the reciprocal of both sides
$ \dfrac{1}{3} \gt \dfrac{1}{4} \\[5ex] $ Teacher: Did you notice what happened?
Student: The inequality sign is reversed.
Teacher: That's right.

(3.) Whenever you take the reciprocal of both sides of an inequality, the inequality is reversed.


Two Notable Notes After Solving Inequalities

(1.) For all inequalities besides the compound inequalities involving "AND"; express your solution such that the variable is always on the $LHS$.
Example: Write $x \lt 3$ rather than $3 \gt x$
For the compound inequalities involving "AND", write the variable in the middle.
Example: Write $3 \lt x \lt 7$

(2.) Always check the solution(s) of any inequality.
Yes, it is possible that it checks out right even though the solution might be wrong (unlike in the case of an equation). However, it is always a good practice to check!

Teacher: And always check with the ....
Student: main inequality or the original inequality
Teacher: Why is that ...(speaking like an American lol)
Student: Because any modified inequality might be incorrect.
Teacher: Good answer!

Symbols and Meanings

Symbols Meaning
(1.) $\gt$ greater than
(2.) $\lt$ less than
(3.) $\ge$ greater than or equal to (also means: "at least", "no less than")
(4.) $\le$ less than or equal to (also means: "at most", "no more than")
(5.) $\{ \}$ braces - used in set notation
(6.) $\{x |...$ $x$ such that...
(7.) $[\:\: ]$ and $(\:\: )$ brackets and parenthesis - used in interval notation
(8.) $[\:\: ]$ closed interval (closed at both ends)
(9.) $(\:\: )$ open interval (open at both ends)
(10.) $[\:\: )$ half-closed half-open interval (closed at $1^{st}$ end, open at $2^{nd}$ end)
(11.) $(\:\: ]$ half-open half-closed interval (open at $1^{st}$ end, closed at $2^{nd}$ end)
(12.) $[c, d]$ closed interval - includes $c$ and $d$
(13.) $(c, d)$ open interval - excludes $c$ and $d$
(14.) $[c, d)$ half-closed half-open interval - includes $c$, excludes $d$
(15.) $(c, d]$ half-open half-closed interval - excludes $c$, includes $d$
(16.) $\-infty$ negative infinity
(17.) $\infty$ positive infinity
(18.) $(-\infty, \infty)$ from negative infinity to positive infinity

Common Mistakes

(1.) Forgot to change(reverse) the inequality after dividing by a negative number.

(2.) Forgot to change(reverse) the inequality after swapping.

(3.) Closed negative infinity.

(4.) Closed positive infinity.

Student: What is that symbol?
Teacher: You mean the eight - upside down or eight - laying down?
Student: ☺☺☺...yes Sir
Teacher: It is ad infinitum (Latin word), the infinity symbol (English word)
Teacher: What is the least number on earth?
Student: Zero?
Teacher: Which one is better - Mr. A who does not have anything and does not owe anything OR Mr. B who has $\$10$ but is owing $\$50$
Student: I get it. Zero is greater than negative forty
Teacher: Correct.
So, what is the least number on earth?
Student: I do not know.
Teacher: Well, there is no least number on earth.
Neither is there a greatest number on earth.
Student: Okay,...so it goes to either negative infinity or positive infinity
Teacher: That is correct!
In that case ...

NOTE: We do not close negative infinity or positive infinity.
We do not ever put a bracket around negative infinity or positive infinity.
We do not close negative infinity because there is no least number on earth.
We do not close positive infinity because there is no greatest number on earth.

Rules of Inequalities

Rules of Inequalities

$c, d, e$ are real numbers

$ (1.)\:\:If\:\: c \lt d \:\:and\:\: d \lt e, \:\:then\:\: c \lt e ...Transitive\:\:Rule \\[3ex] (2.)\:\:If\:\: c \gt d \:\:and\:\: d \gt e, \:\:then\:\: c \gt e ...Transitive\:\:Rule \\[3ex] (3.)\:\:If\:\: c \lt d, \:\:then\:\: d \gt c \\[3ex] (4.)\:\:If\:\: c \gt d, \:\:then\:\: d \lt c \\[3ex] (5.)\:\:If\:\: c \lt d, \:\:then\:\: -c \gt -d \\[3ex] (6.)\:\:If\:\: c \gt d, \:\:then\:\: -c \lt -d \\[3ex] (7.)\:\:If\:\: c \lt d, \:\:then\:\: \dfrac{1}{c} \gt \dfrac{1}{d} \\[5ex] (8.)\:\:If\:\: c \gt d, \:\:then\:\: \dfrac{1}{c} \lt \dfrac{1}{d} \\[5ex] (9.)\:\:If\:\: c \lt d, \:\:then\:\: (c + e) \lt (d + e) \\[3ex] (10.)\:\:If\:\: c \gt d, \:\:then\:\: (c + e) \gt (d + e) \\[3ex] (11.)\:\:If\:\: c \lt d, \:\:then\:\: (c - e) \lt (d - e) \\[3ex] (12.)\:\:If\:\: c \gt d, \:\:then\:\: (c - e) \gt (d - e) \\[3ex] (13.)\:\:If\:\: c \lt d, \:\:and\:\: e \gt 0; \:\:then\:\: ce \lt de \\[3ex] (14.)\:\:If\:\: c \lt d, \:\:and\:\: e \lt 0; \:\:then\:\: ce \gt de \\[3ex] (15.)\:\:If\:\: c \gt d, \:\:and\:\: e \gt 0; \:\:then\:\: ce \gt de \\[3ex] (16.)\:\:If\:\: c \gt d, \:\:and\:\: e \lt 0; \:\:then\:\: ce \lt de \\[3ex] (17.)\:\:If\:\: c \lt d, \:\:and\:\: e \gt 0; \:\:then\:\: \dfrac{c}{e} \lt \dfrac{d}{e} \\[5ex] (18.)\:\:If\:\: c \gt d, \:\:and\:\: e \gt 0; \:\:then\:\: \dfrac{c}{e} \gt \dfrac{d}{e} \\[5ex] (19.)\:\:If\:\: c \lt d, \:\:and\:\: e \lt 0; \:\:then\:\: \dfrac{c}{e} \gt \dfrac{d}{e} \\[5ex] (20.)\:\:If\:\: c \gt d, \:\:and\:\: e \lt 0; \:\:then\:\: \dfrac{c}{e} \lt \dfrac{d}{e} $

Linear Inequalities

A linear inequality is an inequality containing linear expressions.

Student: What is a linear expression?
Teacher: A linear expression is an expression in which the highest exponent of the independent variable in the expression is $1$
Please review the pre-requisite topic: Expressions and Equations

Compound Linear Inequalities

Polynomial Inequalities

A polynomial inequality is an inequality that has a polynomial function.

You can also write that it is an inequality of the forms:
$ f(x) \lt 0 \\[3ex] OR\:\: f(x) \lt some\:\: value/function \\[3ex] OR\:\: f(x) \le 0 \\[3ex] OR\:\: f(x) \le some\:\: value/function \\[3ex] OR\:\: f(x) \gt 0 \\[3ex] OR\:\: f(x) \gt some\:\: value/function \\[3ex] OR\:\: f(x) \ge 0 \\[3ex] OR\:\: f(x) \ge some\:\: value/function \\[3ex] where\:\: f(x) = ax^n + bx^{n - 1} + cx^{n - 2} + ... + z $

Notable Notes for Solving Polynomial Inequalities

(1.) Make sure the $RHS$ (Right Hand Side) is always $0$ before you begin to solve.
This is because $0$ is the only number that "actually" reveals the inequality: greater than, $\gt$; and less than, $\lt$
It is the only number that differentiates positive numbers from negative numbers.
Positive numbers are greater than zero.
Negative numbers are less than zero.

(2.) For "original" (not modified) polynomial inequalities in which $0$ is already on the $RHS$, there is no need to check. Why?
For such polynomial inequalities; we check as we solve. That saves time. ☺☺☺
It is a "check-as-you-solve" kind of thing.

(3.) For all other polynomial inequalities (polynomial inequalities in which we need to modify so that $0$ is on the $RHS$), we have to check our work. And as usual, we have to check with the original/main inequalities.

(4.) For all polynomial inequalities; to get the boundary points, the polynomials on the $LHS$ must be in factored form.
If the polynomial is not in factored form, then we need to factor it in order to get the boundary points.
For the test points, we can use the polynomial "as is" or we can use the factored form (much better).
However, to get the boundary points; it has to be in factored form.

Absolute Value Inequalities

An absolute value inequality is an inequality in which at least one of it's terms has an absolute value.


The absolute value of a term is the magnitude or modulus of that term regardless of sign.
The absolute value of a term say $x$ is denoted by $|x|$
Some resources/calculators represent it as $abs(x)$


Whenever we solve absolute value inequalities, we have to consider two cases.
First Case: The term inside the absolute value is positive.
Second Case: The term inside the absolute value is negative.

And of course, with Mr. C; you have to always Check your solutions

Rational Inequalities

Recall the definitions:

A rational number is any number that can be written as a fraction where the denominator is not equal to zero.
You can also say that a rational number is a ratio of two integers where the denominator is not equal to zero.
A rational number is a number that can be written as: $$\dfrac{c}{d}$$ where $c, d$ are integers and $d \neq 0$
A rational number can be an integer.
It can be a terminating decimal. Why?
It can be a repeating decimal. Why?
It cannot be a non-repeating decimal. Why?
Ask students to tell you what happens if the denominator is zero.

A rational function is a function of the form: $$\dfrac{c(x)}{d(x)}$$ where $c(x), d(x)$ are functions and $d(x) \neq 0$

Based on the prior definitions, ask students to suggest the definition(s) of a rational inequality.

A rational inequality is an inequality containing a rational function.
You can also say that a rational inequality is an inequality of the forms: $$ \dfrac{c(x)}{d(x)} \lt 0 \\[5ex] OR\:\:\: \dfrac{c(x)}{d(x)} \lt some\:\: value/function \\[5ex] OR\:\:\: \dfrac{c(x)}{d(x)} \gt 0 \\[5ex] OR\:\:\: \dfrac{c(x)}{d(x)} \gt some\:\: value/function \\[5ex] OR\:\:\: \dfrac{c(x)}{d(x)} \le 0 \\[5ex] OR\:\:\: \dfrac{c(x)}{d(x)} \le some\:\: value/function \\[5ex] OR\:\:\: \dfrac{c(x)}{d(x)} \ge 0 \\[5ex] OR\:\:\: \dfrac{c(x)}{d(x)} \ge some\:\: value/function \\[5ex] where\:\: c(x), d(x) \:\:are\:\: polynomial\:\: functions\:\: and\:\: d(x) \neq 0 $$


Notable Notes for Solving Rational Inequalities

(1.) Make sure the $RHS$ (Right Hand Side) is always $0$ before you begin to solve.
This is because $0$ is the only number that "actually" reveals the inequality: greater than, $\gt$; and less than, $\lt$
It is the only number that differentiates positive numbers from negative numbers.
Positive numbers are greater than zero.
Negative numbers are less than zero.

(2.) For "original" (not modified) rational inequalities in which $0$ is already on the $RHS$, there is no need to check. Why?
For such rational inequalities; we check as we solve. That saves time. ☺☺☺
It is a "check-as-you-solve" kind of thing.

(3.) For all other rational inequalities (rational inequalities in which we need to modify so that $0$ is on the $RHS$), we have to check our work. And as usual, we have to check with the original/main inequalities.

(4.) For rational inequalities that involves greater than or equal to, $\ge$; and less than or equal to, $\le$; it is important we note the domain of that inequality before writing our solution.

(5.) For all rational inequalities that we solve, we should have:
Only one fraction/rational function on the $LHS$ and
$0$ on the $RHS$
If we have several fractions/rational functions, we should bring all of them to the $LHS$, and we should simplify them to be only one fraction/rational fraction.
We should solve them as fractions. We should never attempt to remove the fractions.
Here is another major problem caused by Inequalities. I informed you earlier that Inequality does not only cause problem in the real-world. It causes problems in Mathematics.
NOTE: Please do not solve rational inequalities by multiplying each term by the $LCD$ (like we did in Rational Equations).
We have to solve the Rational Inequalities as fractions. We will not get rid of the fractions. We just have to like fractions...and we just have to promote equality in our world!!!
One of the easiest ways of solving Rational Equations is to get rid of the fractions by multiplying each term by the $LCD$.
Well, with Rational Inequality...not so. We have to solve them as fractions.
We need to get all the fractions on the $LHS$, and simplify them as one fraction.
We need to have only $0$ on the $RHS$

Then, after solving; we need to check with the original inequality.

(6.) For all rational inequalities; to get the boundary points:
the numerator polynomial (as applicable) and denominator polynomial on the $LHS$ must be in factored form.
For the test points, we can use the rational inequality "as is" or the factored form (much better).
However, to get the boundary points; we have to put both the numerator and the denominator in factored form.

Solve Inequalities Calculator

This calculator will:
(1.) Solve one-variable linear inequalities.
(2.) Solve one-variable quadratic inequalities.
(3.) Solve one-variable polynomial inequalities.
(4.) Solve one-variable absolute value inequalities.
(5.) Solve one-variable rational inequalities.
(6.) Give the answer(s) in the simplest exact forms.
(7.) Graph the real solutions(roots) on a number line.

To see the answer(s) in decimals, click the "Approximate forms" link.
To see the answer(s) in the simplest / exact forms, click the "Exact forms" link.

To use the calculator, please:
(1.) Type the inequality in the textbox (the bigger textbox).
(2.) Type it according to the examples I listed.
(3.) Delete the "default" inequality in the textbox of the calculator.
(4.) Copy and paste the inequality you typed, into the small textbox of the calculator.
(5.) Click the "Submit" button.
(6.) Check to make sure that it is the correct inequality you typed.
(7.) Review the answers.

Solve

Solve Compact Form Inequalities Calculator

This calculator will:
(1.) Solve compact form inequalities.
(2.) Graph the real solutions(roots) on a number line.

To use the calculator, please:
(1.) Type your equation in the textbox (the bigger textbox).
(2.) Type it according to the examples I listed.
(3.) Delete the "default" equation in the textbox of the calculator.
(4.) Copy and paste the equation you typed, into the small textbox of the calculator.
(5.) Type the variable for which you want to isolate.
(5.) Click the "Submit" button.
(6.) Check to make sure that it is the correct equation you typed.
(7.) Review the answer.

Solve



References

Chukwuemeka, S.D (2016, April 30). Samuel Chukwuemeka Tutorials - Math, Science, and Technology. Retrieved from https://www.samuelchukwuemeka.com

Coburn, J., & Coffelt, J. (2014). College Algebra Essentials ($3^{rd}$ ed.). New York: McGraw-Hill

Kaufmann, J., & Schwitters, K. (2011). Algebra for College Students (Revised/Expanded ed.). Belmont, CA: Brooks/Cole, Cengage Learning.

Lial, M., & Hornsby, J. (2012). Beginning and Intermediate Algebra (Revised/Expanded ed.). Boston: Pearson Addison-Wesley.

Sullivan, M., & Sullivan, M. (2017). Algebra & Trigonometry ($7^{th}$ ed.). Boston: Pearson.

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VAST LEARNERS. (n.d). Retrieved from https://www.vastlearners.com/free-jamb-past-questions/