Oggy and Mia are best friends. They race while going for their Cuemath class. Surprisingly, Mia wins!
Now, here is the confusion.
You don't know how fast they ran, but what you know is Mia was faster than Oggy.
Using mathematical expressions you can write it as \(\ M > O\). This shows how fast Mia (M) was.
Mathematically, these expressions come under algebra and are known as inequalities.
In this chapter, you will learn more about inequalities definition, inequalities rules with help of solved inequalities examples.
Lesson Plan
What Are Inequalities?
Olivia is selected in the 12U Softball. How old is Olivia?
You don't know the age of Olivia, because it doesn't say "equals".
But you do know her age should be less than or equal to 12, so it can be written as \(\text{Olivia Age} < 12\)
These are some practical scenarios related to inequalities.
Inequalities Definition
Inequality is an equation. In inequalities, you use less than, greater than, or not equal instead of an equal sign.
 p ≠ q means that p is not equal to q
 p < q means that p is less than q
 p > q means that p is greater than q
 p ≤ q means that p is less than or equal to q
 p ≥ q means that p is greater than or equal to q
Rules of Inequalities
Solving inequalities rules are special. Here are some listed with inequalities examples.
Inequalities Rule 1
When inequalities are linked up you can jump over the middle inequality.
\(\text{If, p < q and q < d, then p < d}\)
\(\text{If, p > q and q > d, then p > d}\)
Example: If Oggy is older than Mia and Mia is older than Cherry, then Oggy must be older than Cherry.
Inequalities Rule 2
Swapping of numbers p and p results in:
\(\text{If, p > q, then q < p}\)
\(\text{If, p < q, then q < p}\)
Example: Oggy is older than Mia, so Mia is younger than Oggy.
Inequalities Rule 3
Only one of the following is true:
\(\text{p > q or p = q or q > p}\)
Example: Oggy has more money than Mia \(\text{a > b}\)
So, Oggy does not have less money than Mia \(\text{not p<q}\)
Oggy does not have the same amount of money as Mia \(\text{not p=q}\)
Inequalities Rule 4
Adding of the number d to both sides of inequality
If \(\text{p < q, then p + d < q + d}\)
Example: Oggy has less money than Mia.
If both Oggy and Mia get $5 more, then Oggy will still have less money than Mia.
Likewise:
If \(\text{p < q, then p − d < q − d}\)
If \(\text{p > q, then p + d > q + d}\), and
If \(\text{p > q, then p − d > q − d}\)
So, addition and subtraction of the same value to both p and q will not change the inequality.
Inequalities Rule 5
If you multiply numbers p and q by a positive number, there is no change in inequality.
If you multiply both p and q by a negative number, the inequality swaps.
p<q becomes q<p after multiplying by (2)
Here are the rules:
If \(\text{p < q, and d is positive, then pd < qd}\)
If \(\text{p < q, and d is negative, then pd > qd}\) (inequality swaps)
Positive case example:
Oggy's score of 5 is lower than Mia's score of 9 \(\text{p < q}\).
If Oggy and Mia double their scores \(\times 2\), Oggy's score will still be lower than Mia's score \(\text{2p < 2q}\).
If the scores turn minuses, then scores will be \(\text{−p > −q}\).
Inequalities Rule 6
Putting minuses in front of p and q changes the direction of the inequality.
If \(\text{p < q then −p > −q}\)
If \(\text{p > q, then −p < −q}\)
It is the same as multiplying by (1) and changes direction.
Inequalities Rule 7
Taking the reciprocal \(\frac{1}{value}\) of both p and q changes the direction of the inequality.
When a and b are both positive or both negative:
If, \(\ p < q, then \frac{1}{p} > \frac{1}{q}\)
If \(\ p > q, then \frac{1}{p} < \frac{1}{q}\)
Inequalities Rule 8
A square of a number is always greater than or equal to zero \(\ p^2\geq 0\)
Example: \(\begin{align}4^2 &= 16\\\ −4^2 &= 16\\\ 0^4 &= 0\end{align}\)
Inequalities Rule 9
Taking a square root will not change the inequality.
If \(\ p\leq q, then\sqrt{a}\leq\sqrt{b}, (for\ p,q\geq 0)\)
Example:
\(\ p=2, q=7\)
\(\ 4\leq 9, \sqrt{4}\leq \sqrt{9}\)
 PEMDAS and BODMAS play a crucial role in solving inequalities.

If the number is negative on either side of a sign (not both) the direction stays the same:
a) If p < q then 1/p < 1/q
b) If p > q then 1/p > 1/q
How to Solve Inequalities?
Jack and Ross play for the same soccer team.
Last Saturday Jack scored 3 more goals than Ross, but together they scored less than 9 goals.
What are the possible number of goals Jack scored?
How do you solve this inequality word problem?
Solution: Break the solution into two parts:
Turn the English into Algebra and use algebraic expressions to solve the word problem.
Let the number of goals Jack scored: J
Let the number of goals Ross scored: R
Jack scored 3 more goals than Ross, so: J = R + 3
Together they scored less than 9 goals: R + J < 9
J = R + 3, so: R + (R + 3) < 9
= 2 R + 3 < 9
Subtract 3 from both sides:
2R < 9 − 3
2R < 6
Divide both sides by 2:
S < 3
Ross scored less than 3 goals, which means that Ross could have scored 0, 1 or 2 goals.
Jack scored 3 more goals than Ross did, so Jack could have scored 3, 4, or 5 goals.
So,
When R = 0, then J = 3 and R + J = 3, and 3 < 9 is correct
When R = 1, then J = 4 and R + J = 5, and 5 < 9 is correct
When R = 2, then J = 5 and R + J = 7, and 7 < 9 is correct
(But when R = 3, then J = 6 and R + J = 9, and 9 < 9 is incorrect)

Imagine 8 cats sitting on a porch. In the group, there are more females than males cats.
How many female cats can there be? Show some tricks with inequalities.
Graphing Inequalities
How to Graph Inequalities?
First, you will have to plot the "equals" line and then, shade the appropriate area.
There are three steps:
 Write the equation such as "y" is on the left and everything else on the right.
 Plot the "y=" line (draw a solid line for y≤ or y≥, and a dashed line for y< or y>)
 Shade the region above the line for a "greater than" (y> or y≥) or below the line for a "less than" (y< or y≤).
Let us try some example:
This is a graph of a linear inequality: y ≤ x + 4
You can see, y = x + 4 line and the shaded area is where y is less than or equal to x + 4
List of notes to help you in solving inequalities:

Inequalities can be solved by adding, subtracting, multiplying, or dividing both sides by the same number.

Dividing or multiplying both sides by negative numbers will alter the inequality's direction.

Do not multiply or divide by a variable.

Linear inequalities are like linear equations (such as 3y = 4x+5) but, they will have inequalities like >, <, ≤, or ≥ rather than an =.
Solved Examples
Example 1 
After reading the chapter inequalities, Gloria observed that for the following inequalities the value of the variables x and y is less than or equal to an integer. Is she correct?
Solve the following equation to check Gloria's assumptions.
a) \(\ 3x  4 < 5\)
b) \(\ 3(5  y) ≥ 9\)
Solution
a) \(\ 3x  4 < 5\)
Add 4 to both sides
⇒ \(\ 3x  4 + 4 < 5 + 4\)
⇒ \(\ 3x < 9\)
Divide both sides by 3
⇒ \(\frac{3x}{3} < \frac{12}{3}\)
⇒ \(\ x < 3\)
b) \(\ 3(5  y) ≥ 9\)
Divide both sides by 3
⇒ \(\frac{3(5  y)}{3} ≥ \frac{9}{3}\)
⇒ \(\ 5  y ≥ 3\)
Subtract 4 from both sides
⇒ \(\ 5  y  5 ≥ 3  5\)
⇒ \(\ y ≥ 2\)
Divide both sides by 1 to change y to y and remember to reverse the inequality sign:
⇒ \(\frac{y}{1} ≤ \frac{2}{1}\)
⇒ \(\ y ≤ 2\)
\(\therefore\) a) x < 3, b) y ≤ 2 
Example 2 
Turn the statement form into a linear inequality equation.
2 is added to a number x. Then the result is multiplied by 9, The final answer is greater than 28
Solution
Start with x.
Add 2 ⇒ x + 2
Multiply by 9 ⇒ 9(x + 2).
The final answer is greater than 28 ⇒ 9(x + 2) > 28
\(\therefore\) 9(x + 2) > 28 
Example 3 
Lisa works as a parttime online tutor for $2 per hour with a fixed incentive of $1 She is allowed to work for a maximum of 3 hours per week.
Use a graphical method to solve the system of linear inequalities.
\(\ y = 2x + 1\)
\(\ y ≤ 3\)
Solution
1. Draw the line \(\ y = 2x + 1\)
2. Draw the line \(\ y = 3\)
x ≤ 1 and \(\ y ≤ 3\)
Interactive Questions
Here are a few activities for you to practice.
Select/Type your answer and click the "Check Answer" button to see the result.
Let's Summarize
We hope you enjoyed learning about inequalities with the simulations and practice questions. Now, you will be able to easily solve practice problems on inequalities rules, inequalities definition, and inequalities examples.
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Frequently Asked Questions (FAQs)
1. How do you solve inequalities on a number line?
To solve inequalities on a number line see the example below.
2. What happens when you square an inequality?
A square of a number is always greater than or equal to zero \(\ p^2\geq 0\)
Example:
\(\begin{align}4^2 &= 16\\\ −4^2 &= 16\\\ 0^4 &= 0\end{align}\)
3. How do you find the range of inequality?
You can find the range of values of x, by solving the inequality by considering it as a normal linear equation.