Inequalities
In Mathematics, equations are not always about being balanced of both sides with an 'equal to' symbol. Sometimes it can be about 'not an equal to' relationship like something is greater than the other or less than. In mathematics, inequality refers to a relationship that makes a nonequal comparison between two numbers or other mathematical expressions. These mathematical expressions come under algebra and are called inequalities.
What Are Inequalities?
The mathematical expressions in which both sides are not equal are called inequalities. In inequality, unlike in equations, we compare two values.The equal to sign in between is replaced by less than, greater than or not equal to sign.
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 Olivia's Age < 12. This is a practical scenario related to inequalities.
 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
The rules of inequalities are special. Here are some listed with inequalities examples.
Inequalities Rule 1
When inequalities are linked up you can jump over the middle inequality.
 If, p < q and q < d, then p < d
 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:
 If, p > q, then q < p
 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: p > q or p = q or q > p
Example: Oggy has more money than Mia (a > b). So, Oggy does not have less money than Mia (not p<q). Oggy does not have the same amount of money as Mia (not p=q)
Inequalities Rule 4
Adding of the number d to both sides of inequality If 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 p < q, then p − d < q − d
 If p > q, then p + d > q + d, and
 If 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 p < q, and d is positive, then pd < qd
 If 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 (p < q). If Oggy and Mia double their scores '×2', Oggy's score will still be lower than Mia's score, 2p < 2q. If the scores turn minuses, then scores will be −p > −q.
Inequalities Rule 6
Putting minuses in front of p and q changes the direction of the inequality.
 If p < q then −p > −q
 If p > q, then −p < −q
 It is the same as multiplying by (1) and changes direction.
Inequalities Rule 7
Taking the reciprocal 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, then1/p > 1/q
 If p > q, then1/p < 1/q
Inequalities Rule 8
A square of a number is always greater than or equal to zero \(\ p^2\geq 0\)
Example: (4)^{2}= 16, (−4)^{2} = 16, (0)^{2} = 0
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
\(\ 2\leq 7, \sqrt{2}\leq \sqrt{7}\)
How to Solve Inequalities?
To solve an inequality, we can use the following steps:
 Step 1: Eliminate fractions(multiplying all terms by the least common denominator of all fractions).
 Step 2: Simplify(combine like terms on each side of the inequality).
 Step 3: Add or subtract quantities(unknown on one side and the numbers on the other).
Example: 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 is the possible number of goals Jack scored? How do you solve this inequality word problem?
Solution: Break the solution into two parts: Turn the statements into 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 and 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 note that when R = 3, then J = 6 and R + J = 9, and 9 < 9 is incorrect)
Graphing Inequalities
For graph inequalities, 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.
Tips and Tricks
 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
Thinking Out Of the Box!
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.
Important Notes
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 =.
Topics Related to Inequalities
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 ⇒ 3x/3 <9/3 ⇒ 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. 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
FAQs on Inequalities
How Do you Solve Inequalities On A Number Line?
To plot an inequality, such as x>3, on a number line,
 Step 1: Draw a circle over the number (e.g., 3).
 Step 2: Check if the sign includes equal to (≥ or ≤) or not. If equal to sign is there along with > or <, then fill in the circle otherwise leave the circle unfilled.
 Step 3: On the number line, extend the line from 3(after encircling it) to show it is greater than or equal to 3.
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: (4)^{2} = 16, (−4)^{2} = 16, (0)^{2} = 0
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.
What Are the 5 Inequality Symbols?
The 5 inequality symbols are less than (<), greater than (>), less than or equal (≤), greater than or equal (≥), and the not equal symbol (≠).
How Do you Tell If It's An Inequality?
Equations and inequalities are mathematical sentences formed by relating two expressions to each other. In an equation, the two expressions are supposed to be equal and shown by the symbol =. Whereas in inequality, the two expressions are not necessarily equal and are indicated by the symbols: >, <, ≤ or ≥.