If there is one prayer that you should

- Samuel Dominic Chukwuemeka
**pray/sing** every day and every hour, it is the
LORD's prayer (Our FATHER in Heaven prayer)

It is the **most powerful prayer**.
A **pure heart**, a **clean mind**, and a **clear conscience** is necessary for it.

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

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

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.

(1.) Use of prior knowledge

(2.) Critical Thinking

(3.) Interdisciplinary connections/applications

(4.) Technology

(5.) Active participation through direct questioning

(6.) Research

**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

*
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...

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.
*

$2 + 3 = 5$

$5 = 2 + 3$

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.

$(2 + 3) \lt 6$

$6 \gt (2 + 3)$

Student: The inequality sign is reversed.

Teacher: That's right.

$3 = 3$

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

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

$3 \lt 4$

Take the reciprocal of both sides

$ \dfrac{1}{3} \gt \dfrac{1}{4} \\[5ex] $

Student: The inequality sign is reversed.

Teacher: That's right.

**
(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 | 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 1st end, open at 2nd end) |

(11.) $(\:\: ]$ | half-open half-closed interval (open at 1st end, closed at 2nd 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 |

(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 *

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.

$c, d, e$ 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} $

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
*

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
$

(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.

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**

__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:
$$
\dfrac{c(x)}{d(x)} \lt 0 \\[5ex]
OR\:\:\: \dfrac{c(x)}{d(x)} \lt some\:\: value/function \\[5ex]
OR\:\:\: \dfrac{c(x)}{d(x)} \gt 0 \\[5ex]
OR\:\:\: \dfrac{c(x)}{d(x)} \gt some\:\: value/function \\[5ex]
OR\:\:\: \dfrac{c(x)}{d(x)} \le 0 \\[5ex]
OR\:\:\: \dfrac{c(x)}{d(x)} \le some\:\: value/function \\[5ex]
OR\:\:\: \dfrac{c(x)}{d(x)} \ge 0 \\[5ex]
OR\:\:\: \dfrac{c(x)}{d(x)} \ge some\:\: value/function \\[5ex]
where\:\: c(x), d(x) \:\:are\:\: polynomial\:\: functions\:\: and\:\: d(x) \neq 0
$$

(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

(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.

__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.

- Using the Solve Inequalities Calculator
**Linear Inequality:**Type: $x + 18 \lt -3$__as__x + 18 "then-select the less-than symbol" -3**Linear Inequality:**Type: $3x - 1 \ge 5$__as__3 * x - 1 "then-select the greater-than-or-equal-to symbol" 5

Solve

__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.

- Using the Solve Compact Form Inequalities Calculator

Solve

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