An inequality is a mathematical statement that two quantities are not equal in value. You can use inequality symbols to show that one quantity is greater than or less than another quantity. The following outlines the different inequality symbols and their meanings:
Just like with equations, you can add or subtract a value on both sides of an inequality and keep the inequality statement true.
EXAMPLE
If you have the inequality 3 is less than 5, and you add 2 to both sides, you have 3 plus 2 is less than 5 plus 2. This simplifies to 5 is less than 7, which is still a true statement.EXAMPLE
If you have the inequality 6 is greater than or equal to 4, and you multiply by 3 on both sides, you have 3 times 6 is greater than or equal to 3 times 4. This simplifies to 18 is greater than or equal to 12, which is still true.EXAMPLE
If you have the inequality 6 is greater than or equal to 4, and you multiply by a negative 3 on both sides, you have negative 3 times 6 is greater than or equal to negative 3 times 4, which simplifies to negative 18 is greater than or equal to negative 12. Clearly, this is an untrue statement until you flip the sign. When you flip the sign, you now have negative 18 is less than or equal to negative 12, which is a true statement.You may recall that in solving an equation, you isolate the variable using inverse operations. To isolate the variable, inverse operations are used to get all terms involving the variable on one side of the equation and all other terms to the other side of the equation.
The process for solving an inequality follows the same rules. You use inverse operations to isolate the variable, and what is done on one side of the inequality must be done on the other.
IN CONTEXT
Suppose Marcus has accepted a job selling cell phones. He will be paid $1,500 plus 15% of his sales each month. He needs to earn at least $2,430 per month to pay his bills. What amount of sales does Marcus need each month to be able to pay his bills?
Let the variable x represent the unknown quantity, which is the amount of sales Marcus needs each month. You know that 1,500 is the amount of Marcus's base monthly salary. You also know that Marcus earns an additional 15%, or 0.15, of his total sales each month. The combination of his base salary and his percentage of total sales has to be greater than or equal to 2,430, the amount of his bills. Therefore the inequality becomes:
Our inequality
Start by subtracting 1500 on both sides to isolate your xvariable.
Next, divide both sides by 0.15
Our solution
You can show this solution on a number line and use it to check your answer. If you pick a value in the highlighted range below, it should satisfy your original inequality. For example, if you pick the value 6,500 and substitute it back into your inequality, you have:
Simplify the expression, starting with multiplication and moving on to addition, which gives you the inequality 2,475 is greater than or equal to 2,430, which is a true statement.

Our inequality  
 

Start by isolating the valuable on one side. Subtract 1500t from both sides of the inequality.  
 

Next, subtract 25000 from both sides of the inequality  
 

Finally, divide by negative 500  
 

Since we are dividing by a negative number, we must flip the inequality sign. t is greater than 12 is our answer. Jaime will make more than Wei after 12 years. 
Again, you can show the solution on a number line, and if you pick a value in the highlighted range it should satisfy your original inequality. For example, if you pick the value 15 and substitute it back into your original inequality for t, you have:
Source: This work is adapted from Sophia author Colleen Atakpu.