Introduction
 An inequality is a mathematical sentence that makes a nonequal comparison between two mathematical expressions.
 There are four inequality operators, as shown in the table below:
Operator  Symbol 
Less than  < 
Graeter than  > 
Less than or equal to  $\le $ 
Greater than or equal to  $\ge $ 
 Examples of inequality include: $2x>4$, $5y<20$, $4m2\ge 6$, etc.
Solving simple linear inequalities
 Inequalities can be solved using the rules mentioned below.
 Addition Rule:

 Adding the same number to each side of an inequality produces an equivalent inequality wherein the sign of inequality remains unaffected.
 Subtraction Rule:

 Subtracting the same number from both sides of an inequality results in an equivalent inequality wherein the sign of inequality remains unaffected.
 Multiplication Rule:

 Multiplying both sides of an inequality by a positive number results in an equivalent inequality wherein the sign of inequality remains unaffected.
 Multiplying both sides of an inequality by a negative number results in a new inequality wherein the sign of inequality is reversed.
 Division Rule:

 Dividing both sides of an inequality by a positive number results in an equivalent inequality wherein the sign of inequality remains unaffected.
 Dividing both sides of an inequality by a negative number results in a new inequality wherein the sign of inequality is reversed.
Graphical Representation of Linear Inequalities in One Variable
 Linear inequalities in one variable can be visualized on a number line.
 Plotting linear inequalities in one variable involves the following steps:

 First, solve the linear inequality to find the variable's value that satisfies the inequality.
 Locate and mark the solution on the number line.
 We use an open dot for strict inequalities to indicate that the point is not part of the solution.
 We use a closed dot for slack inequalities to indicate that the point is part of the solution.
Solved Examples
Question 1: Solve the following inequality.
$m2\le 5$
Solution: $m2\le 5$
Applying addition rule
$m2+2\le 5+2$
$m\le 7$
Question 2: Solve the following inequality.
$x+3>1$
Solution: $x+3>1$
Applying subtraction rule
$x+33>13$
$x>2$
Question 3: Solve the following inequality.
$\frac{x}{2}\ge 4$
Solution: $\frac{x}{2}\ge 4$
Apply multiplication rule (by a positive number)
$\frac{x}{2}\times 2\ge 4\times 2$
$x\ge 8$
Question 4: Solve the following inequality.
$\frac{m}{4}\le 3$
Solution: $\frac{m}{4}\le 3$
Apply multiplication (by a negative number) rule
$\frac{m}{4}\times \left(4\right)\ge 3\times \left(4\right)$ [Note that the inequality sign reversed due to multiplication by a negative number]
$m\ge 12$
Question 5: Solve the following equation.
$7x<21$
Solution: $7x<21$
Apply the division rule (by a positive number)
$\frac{7x}{7}<\frac{21}{7}$
$x<3$
Question 6: Solve the following inequality.
$2x\ge 8$
Solution: $2x\ge 8$
Apply division (by a negative number) rule
$\frac{2x}{2}\ge \frac{8}{2}$
$x\ge 4$
Question 7: What inequality does the number line shown below represents?
Solution: An open circle is used to signify "less than (<)" or "greater than (>)" type inequality on the number line. The given figure represents $x<10$.
Question 8: What inequality does the number line shown below represents?
Solution: A closed circle denotes "less than equal to ($\le $)" or "more than equal to ($\ge $)" type inequality on the number line. The given figure represents $x\ge 1$.
Cheat Sheet
 If two mathematical expressions are separated by symbols such as >, <, $\ge $, $\le $, it represents an inequality.
 To solve an inequality, we can apply the rules of addition, subtraction, multiplication, and division.
 Adding the same quantity to both sides of an inequality doesn't change the direction of the inequality sign.
 Likewise, subtracting the same quantity from both sides of an inequality doesn't change the direction of the inequality sign.
 Multiplying both sides of an inequality by a positive number leaves the inequality sign unchanged. In contrast, multiplication by a negative number reverses the direction of inequality.
 Similarly, dividing both sides of an inequality by a positive number leaves the inequality sign unchanged. At the same time, division by a negative number reverses the direction of the inequality.
 Linear inequality in one variable is plotted on a number line. An open circle is used to signify "less than (<)" or "greater than (>)", while a closed circle denotes "less than equal to ($\le $)" or "more than equal to ($\ge $)" type inequality on the number line.
Blunder Areas
 When solving inequalities, one must not forget to reverse the inequality sign when multiplying or dividing both sides by negative numbers.
 Abhishek Tiwari
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