All About Inequalities Definition

In mathematics, inequalities are defined as those expressions in which both sides are not equal. One side is greater than, less than or can be greater than equals to or less than equals to from another side. In case of inequalities, = (equals to) sign is replaced by < (less than), > (greater than), ≥ (greater than equals to), ≤ (less than equals to), ≠(not equals to).

Polynomial inequalities, rational inequalities, and absolute value inequalities are the different types of inequalities that exist in mathematics.

Linear inequalities

As you very well know, linear equations means those equations which contain only single power.

Same as this, linear inequalities are those in which all terms are present and have single power only. These inequalities also can be polynomial, rational, or absolute value inequalities.

Methods for solving linear inequalities

The steps which are given below will help you to solve inequality problems.

Step – 1: Firstly, assume the inequality as an equation and also write it in the form of an equation.

Step – 2: Solve the equation with the method of solving equations and note that you have to solve the equation for more than one value.

Step – 3: Form the number line and represent all the obtained values on that number line. 

Step – 4: Use open and filled circles for excluding and including values on the number line respectively.

Step – 5: Find the interval that you are getting by the intersection of values obtained on the number line.

Step – 6: Now, you have an interval, which is the solution to the inequality. To validate your answer, you can take any random value from the interval, and put it in the given inequality, if the value satisfies the inequality then your answer is correct.

Note The interval that satisfies the inequality that will be the solution to the given inequality.

Graphing of Linear Inequality

When you draw the graph for any linear equation, you have noticed that you have got the line which consists of the solution of the linear equation. But in the case of linear inequality when you plot it on a graph, you get linear straight lines, but the solution of a linear inequality is the area of the coordinate plane which is surrounded by lines.

The linear inequalities divide the coordinate plane into two parts and that is done by a borderline. This line itself belongs to that function. One part consists of all the solutions that inequalities have and another part does not contain any solution. That one part itself is that area that is the solution of the inequality equation.

Steps For graphing Linear inequalities

1. Rearrange the equation such that y is on the left-hand side and the rest of the art on the right hand – side.
2. Now, plot the inequalities on a graph firstly by finding their solution.
3. Draw a solid line for y≤ or y≥ and a dashed line for y< or y>.
4. Shade the line according to inequality, you have to draw above the line for a “greater than” (y> or y≥) and below the line for a “less than” (y< or y≤).

Examples of Linear inequality

Example 1: Find the solution of the linear inequality, 3x + 6 > 4, and plot it on a number line.

Solution: We can solve the given linear inequality by the above-given steps.

Step – 1: 3x + 6 > 4

Step – 2: Subtract 6 on both sides in order to keep variable terms on one side and constant terms on another side.

3x + 6 – 6 > 4 – 6

Step – 3: 3x > -2

Step – 4: Divide by 3 both sides, x > -⅔

Example – 2: Which of the following is the solution for the given linear inequality?

4x – 3 > 2x + 5

A} x <  -4

B} x > 4 

C} x < 4

D} x > -4

 Solution: Step – 1: Isolate the variable and the constant terms on different sides

Step – 2: Subtract ‘ 2x’ from both side,

4x – 3 – 2x > 2x + 5 – 2x

Step – 3: 2x – 3 > 5

Step –  4: Add 3 both side,

2x – 3 + 3 > 5 + 3

Step – 5: 2x > 8

Step – 6: Divide by 2 both side,

x > 4

Important facts to keep in mind before solving the problems of linear inequalities-

• Linear equalities have 1 as the highest power.
• ‘Less than’ and ‘greater than’ are the true inequalities. 

Conclusion

In this article, you have studied linear inequalities, their basic concepts and examples of linear inequalities. It will help you in learning the basics of linear inequalities. 

Inequalities are defined as those expressions in which both sides are not equal. One side is greater than, less than or can be greater than equals to or less than equals to from another side.