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Inequalities
In Mathematics, equations are not always about being balanced on 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.
Let us learn the rules of inequalities, and how to solve and graph them.
What is an Inequality?
Inequalities are the mathematical expressions in which both sides are not equal. In inequality, unlike in equations, we compare two values. The equal sign in between is replaced by less than (or less than or equal to), greater than (or greater than or equal to), 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.
Inequality Meaning
The meaning of inequality is to say that two things are NOT equal. One of the things may be less than, greater than, less than or equal to, or greater than or equal to the other things.
 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
There are different types of inequalities. Some of the important inequalities are:
 Polynomial inequalities
 Absolute value inequalities
 Rational inequalities
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 q 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
Adding 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, the addition and subtraction of the same value to both p and q will not change the inequality.
Inequalities Rule 4
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 5
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 6
Taking the reciprocal 1/value of both p and q changes the direction of the inequality. When p and q are both positive or both negative:
 If, p < q, then 1/p > 1/q
 If p > q, then 1/p < 1/q
Inequalities Rule 7
A square of a number is always greater than or equal to zero p^{2 }≥ 0.
Example: (4)^{2}= 16, (−4)^{2} = 16, (0)^{2} = 0
Inequalities Rule 8
Taking a square root will not change the inequality. If p ≤ q, then √p ≤ √q (for p, q ≥ 0).
Example:
p=2, q=7
2 ≤ 7, then √2 ≤ √7
The rules of inequalities are summarized in the following table.
Operation Applied While Solving Inequalities  Sign change? 

Addition on both sides  No 
Subtraction on both sides  No 
Multiplying or dividing both sides by a positive number  No 
Multiplication or dividing both sides by a negative number  Yes 
Swapping both sides  Yes 
Simplify one side  No 
Solving Inequalities
Here are the steps for solving inequalities:
 Step  1: Write the inequality as an equation.
 Step  2: Solve the equation for one or more values.
 Step  3: Represent all the values on the number line.
 Step  4: Also, represent all excluded values on the number line using open circles.
 Step  5: Identify the intervals.
 Step  6: Take a random number from each interval, substitute it in the inequality and check whether the inequality is satisfied.
 Step  7: Intervals that are satisfied are the solutions.
But for solving simple inequalities (linear), we usually apply algebraic operations like addition, subtraction, multiplication, and division. Consider the following example:
2x + 3 > 3x + 4
Subtracting 3x and 3 from both sides,
2x  3x > 4  3
x > 1
Multiplying both sides by 1,
x < 1
Notice that we have changed the ">" symbol into "<" symbol. Why? This is because we have multiplied both sides of the inequality by a negative number. The process of solving inequalities mentioned above works for a simple linear inequality. But to solve any other complex inequality, we have to use the following process.
Let us use this procedure to solve inequalities of different types.
Graphing Inequalities
While graphing inequalities, we have to keep the following things in mind.
 If the endpoint is included (i.e., in case of ≤ or ≥) use a closed circle.
 If the endpoint is NOT included (i.e., in case of < or >), use an open circle.
 Use open circle at either ∞ or ∞.
 Draw a line from the endpoint that extends to the right side if the variable is greater than the number.
 Draw a line from the endpoint that extends to the left side if the variable is lesser than the number.
Writing Inequalities in Interval Notation
While writing the solution of an inequality in the interval notation, we have to keep the following things in mind.
 If the endpoint is included (i.e., in case of ≤ or ≥) use the closed brackets '[' or ']'
 If the endpoint is not included (i.e., in case of < or >), use the open brackets '(' or ')'
 Use always open bracket at either ∞ or ∞.
Here are some examples to understand the same:
Inequality  Interval 

x < 2  (∞, 2) 
x > 2  (2, ∞) 
x ≤ 2  (∞, 2] 
x ≥ 2  [2, ∞) 
2 < x ≤ 6  (2, 6] 
Graphing Inequalities with Two Variables
For graphing inequalities with two variables, 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 (in yellow) is where y is less than or equal to x + 4. Let us now see how to solve different types of inequalities and how to graph the solution in each case.
Solving Polynomial Inequalities
The polynomial inequalities are inequalities that can be expressed as a polynomial on one side and 0 on the other side of the inequality. There are different types of polynomial inequalities but the important ones are:
 Linear Inequalities
 Quadratic Inequalities
Solving Linear Inequalities
A linear inequality is an inequality that can be expressed with a linear expression on one side and a 0 on the other side. Solving linear inequalities is as same as solving linear equations, but just the rules of solving inequalities (that was explained before) should be taken care of. Let us see some examples.
Solving One Step Inequalities
Consider an inequality 2x < 6 (which is a linear inequality with one variable). To solve this, just one step is sufficient which is dividing both sides by 2. Then we get x < 3. Therefore, the solution of the inequality is x < 3 (or) (∞, 3).
Solving Two Step Inequalities
Consider an inequality 2x + 3 > 6. To solve this, we need two steps. The first step is subtracting 3 from both sides, which gives 2x > 3. Then we need to divide both sides by 2 and it results in x < 3/2 (note that we have changed the sign of the inequality). So the solution of the inequality is x < 3/2 (or) (∞, 3/2).
Solving Compound Inequalities
Compound inequalities refer to the set of inequalities with either "and" or "or" in between them. For solving inequalities, in this case, just solve each inequality independently and then find the final solution according to the following rules:
 The final solution is the intersection of the solutions of the independent inequalities if we have "and" between them.
 The final solution is the union of the solutions of the independent inequalities if we have "or" between them.
Example: Solve the compound inequality 2x + 3 < 5 and x + 6 < 3.
Solution:
By first inequality:
2x + 3 < 5
2x < 8
x < 4
By second inequality,
x + 6 < 3
x < 3
Since we have "and" between them, we have to find the intersection of the sets x < 4 and x < 3. A number line may be helpful in this case. Then the final solution is:
x < 3 (or) (∞, 3).
Solving Quadratic Inequalities
A quadratic inequality involves a quadratic expression in it. Here is the process of solving quadratic inequalities. The process is explained with an example where we are going to solve the inequality x^{2}  4x  5 ≥ 0.
 Step 1: Write the inequality as equation.
x^{2}  4x  5 = 0  Step 2: Solve the equation. Here we can use any process of solving quadratic equations.
Then (x  5) (x + 1) = 0
x = 5; x = 1.  Step 3: Represent the solutions from Step 2 on the number line and identify the intervals. Take care of open circles and closed circles.
Since we have '≥' here (that involves '='), we use closed circles at both 5 and 1.
 Step 4: Take a random number from each interval and check whether the inequality is true for that number.
Interval Random Number Checking the Inequality (∞, 1] x = 2 (2)^{2}  4(2)  5 ≥ 0
7 ≥ 0, true[1, 5] x = 0 (0)^{2}  4(0)  5 ≥ 0
5 ≥ 0, false[5, ∞) x = 6 (6)^{2}  4(6)  5 ≥ 0
7 ≥ 0, true  Step 5: The inequalities with "true" from the above table are solutions.
Therefore, the solutions of the quadratic inequality x^{2}  4x  5 ≥ 0 is (∞, 1] U [5, ∞).
We can use the same process for solving cubic inequalities, biquadratic inequalities, etc.
Solving Absolute Value Inequalities
An absolute value inequality includes an algebraic expression inside the absolute value sign. Here is the process of solving absolute value inequalities where the process is explained with an example of solving an absolute value inequality x + 3 ≤ 2. If you want to learn different methods of solving absolute value inequalities, click here.
 Step 1: Consider the absolute value inequality as equation.
x + 3 = 2  Step 2: Solve the equation.
x + 3 = ±2
x + 3 = 2; x + 3 = 2
x = 1; x = 5  Step 3: Represent the numbers from Step 2 on the number line and identify the intervals.
Since '≤' involves "=", we use closed brackets at 1 and 5.
 Step 4: Take a random number for testing from each of the above intervals and check whether the given inequality gets satisfied.
Interval Random Number Checking the Inequality (∞, 5] 6 6 + 3 ≤ 2
3 ≤ 2, false[5, 1] 3 3 + 3 ≤ 2
0 ≤ 2, true[1, ∞) 0 0 + 3 ≤ 2
3 ≤ 2, false  Step 5: The intervals that satisfied the inequality are the solution intervals.
Therefore, the solution of the absolute value inequality x + 3 ≤ 2 is [5, 1].
Solving Rational Inequalities
Rational inequalities are inequalities that involve rational expressions (fractions with variables). To solve the rational inequalities (inequalities with fractions), we just use the same procedure as other inequalities but we have to take care of the excluded points. For example, while solving the rational inequality (x + 2) / (x  2) < 3, we should note that the rational expression (x + 2) / (x  2) is NOT defined at x = 2 (set the denominator x  2 = 0 ⇒x = 2). Let us solve this inequality step by step.
 Step 1: Consider the inequality as the equation.
(x + 2) / (x  2) = 3  Step 2: Solve the equation.
x + 2 = 3(x  2)
x + 2 = 3x  6
2x = 8
x = 4  Step 3: Represent the number(s) from the above step and the excluding values on the number line. Note that the open circle/closed circle in case of the numbers from the above step depends on the given inequality whereas we always get an open circle at the excluded number as its name suggests.
Since the given inequality has no "=" sign in it, we just get an open circle at 4, and since 2 is the excluded value we get an open circle at it.
 Step 4: Let us take some random numbers from each of the above intervals and test the given inequality
Interval Random Number Checking the inequality (∞, 2) 0 (0 + 2) / (0  2) < 3
1 < 3, true(2, 4) 3 (3 + 2) / (3  2) < 3
5 < 3, false(4, ∞) 5 (5 + 2) / (5  2) < 3
2.3 < 3, true
 Step 5: The intervals that have come up with "true" in Step 4 are the solutions.
Therefore, the solution of the rational inequality (x + 2) / (x  2) < 3 is (∞, 2) U (4, ∞).
Important Notes on Inequalities:
Here are the notes about inequalities:
 If we have strictly less than or strictly greater than symbol, then we never get any closed interval in the solution.
 We always get open intervals at ∞ or ∞ symbols because they are NOT numbers to include.
 Write open intervals always at excluded values when solving rational inequalities.
 Excluded values should be taken care of only in case of rational inequalities.
☛ Related Topics:
Inequalities Examples

Example 1: Using the techniques of solving inequalities, solve: 19 < 3x + 2 ≤ 17 and write the answer in the interval notation.
Solution:
Given that 19 < 3x + 2 ≤ 17.
This is a compound inequality.
Subtracting 2 from all sides,
21 < 3x ≤ 15
Dividing all sides by 3,
7 < x ≤ 5
Answer: The solution is (7, 5].

Example 2: While solving inequalities, explain why each of the following statements is incorrect. Also, correct them.
a) 2x < 5 ⇒ x > 5/2
b) x > 3 ⇒ x ∈ [3, ∞)
c) x > 7 ⇒ x > 7.Solution:
a) 2x < 5. Here, when we divide both sides by 2, which is a positive number, the sign does not change. So the correct inequality is x < 5/2.
b) x > 3. It does not include an equal to symbol. So 3 should NOT be included in the interval. So the correct interval is (3, ∞).
c) x > 7. When we divide both sides by 1, a negative number, the sign should change. So the correct inequality is x < 7.
Answer: The corrected ones are a) x < 5/2; b) x ∈ (3, ∞); c) x < 7.

Example 3: Solve the inequality x^{2}  7x + 10 < 0.
Solution:
First, solve the equation x^{2}  7x + 10 = 0.
(x  2) (x  5) = 0.
x = 2, x = 5.
If we represent these numbers on the number line, we get the following intervals: (∞, 2), (2, 5), and (5, ∞).
Let us take some random numbers from each interval to test the given quadratic inequality.
Interval Random Number Checking the Inequality (∞, 2) 0 0^{2}  7(0) + 10 < 0
10 < 0, false(2, 5) 3 3^{2}  7(3) + 10 < 0
2 < 0, true(5, ∞) 6 6^{2}  7(6) + 10 < 0
4 < 0, falseTherefore, the only interval that satisfies the inequality is (2, 5).
Answer: The solution is (2, 5).
FAQs on Inequalities
What are Inequalities in Math?
When two or more algebraic expressions are compared using the symbols <, > ≤, or ≥, then they form an inequality. They are the mathematical expressions in which both sides are not equal.
How Do you Solve Inequalities On A Number Line?
To plot an inequality in math, 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.
How to Calculate Inequalities in Math?
To calculate inequalities:
 just make it an equation
 solve it
 mark the zeros on the number line to get intervals
 test the intervals by taking any one number from it against the inequality.
Explain the Process of Solving Inequalities Graphically.
Solving inequalities graphically is possible when we have a system of two inequalities in two variables. In this case, we consider both inequalities as two linear equations and graph them. Then we get two lines. Shade the upper/lower portion of each of the lines that satisfies the inequality. The common portion of both shaded regions is the solution region.
What is the Difference Between Equations and Inequalities?
Here are the differences between equations and inequalities.
Equations  Inequalities 

1. Equations have "=" symbol in it.  1. Inequalities have ">", "<", "≥", or "≤" in it 
2. The number of solutions of an equation depends on the degree of the equation.  2. An inequality may have a single, unique, or no solution. It doesn't depend on the degree. 
3. By applying any operation on both sides, an equation still holds.  3. If we multiply/divide both sides of an inequality by a negative number, the sign changes. 
What Happens When you Square An Inequality?
A square of a number is always greater than or equal to zero p^{2} ≥ 0. Example: (4)^{2} = 16, (−4)^{2} = 16, (0)^{2} = 0
What are the Steps to Calculate Inequalities with Fractions?
Calculating inequalities with fractions is just like solving any other inequality. One easy way of solving such inequalities is to multiply every term on both sides by the LCD of all denominators so that all fractions become integers. For example, to solve (1/2) x + 1 > (3/4) x + 2, multiply both sides by 4. Then we get 2x + 4 > 3x + 8 ⇒ x > 4 ⇒ x < 4.
What are the Steps for Solving Inequalities with Variables on Both Sides?
When an inequality has a variable on both sides, we have to try to isolate the variable. But in this process, flip the inequality sign whenever we are dividing or multiplying both sides by a negative number. Here is an example. 3x  7 < 5x  11 ⇒ 2x < 4 ⇒ x > 2.
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 ≥.
How to Graph the Solution After Solving Inequalities?
After solving inequalities, we can graph the solution keeping the following things in mind.
 Use an open circle at the number if it is not included and use a closed circle if it is included.
 Draw a line to the right side of the number in case of '>' and to the left side of the number in case of '<'.
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