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# Inequality Solver

The inequality solver is used to solve many engineering problems and science application issues. The inequality symbols used to compare numbers and to denote subsets of real numbers. The inequality symbols $<$, $>$, $\geq$, $\leq$ used to compare quantities. To solve an inequality in the variable by finding all values of that variable for which the inequality is true. These values are solutions of the inequality and are said to satisfy the inequality. An inequality is a statement which says two expressions can be true for a set of values assigned to the variable. The inequalities not having the options of ‘equal to’ are called ‘strict inequalities’.

## Method of Solving an Inequality

The method of solving an inequality is exactly the same as solving an equation. All necessary algebraic operations can be performed as you do in case of equations for isolating the variable. There is only one very important difference to be noted, when you reverse the signs on both sides of an inequality, the inequality symbol MUST be reversed. For example if – x > - 1, then x < 1.

### Solved Example

Question: Solve  5x + 2 < 2x - 7

Solution:
Given inequality,
5x + 2 < 2x - 7

Subtract 2 from both sides

=> 5x + 2 - 2 < 2x - 7 - 2

=> 5x < 2x - 9

Subtract 2x from both sides

=> 5x - 2x < 2x - 9 - 2x

=> 3x < - 9

Divide each side by 3

=> $\frac{3x}{3} < \frac{-9}{3}$

=> x < - 3

So, the solution set is all real numbers that are less than 1. But x = -3 is not part of the solution.

## Solving a Linear Inequality

An equation is a statement which says two expressions are true for a particular value of the variable. An inequality says two expressions can be true for a set of values assigned to the variable. Inequality solver helps to solve the values of variables for which inequality is true. The simplest type of inequality to solve is a linear inequality in one variable. Linear inequality also named as one-step linear inequality. An inequality with one or two variables of order 1 is called as ‘linear inequality’, this inequality will have infinite number of solution.

### Solved Example

Question: Solve 3x + 6 > 9

Solution:
Given
3x + 6 > 9

Subtract 6 from each side

=> 3x + 6 - 6 > 9 - 6

=> 3x > 3

divide each side by 3

=> $\frac{3x}{3} > \frac{3}{3}$

=> x > 1

So, the solution set is all real numbers that are greater than 1. But x = 1 is not part of the solution.

An inequality in which the order of the variable is 2, then it is called a quadratic inequality. Like we have two solutions in a quadratic equation we will have two solution sets in case of a quadratic inequality.

For example, consider an inequality x2 – x – 6 > 0.

It can be factorized as (x – 3)(x + 2) > 0.

But you have to use the property of sign of a product and arrive at the correct solutions. A set of inequalities either linear or quadratic in nature is called system of inequalities. One must be correctly figure out the solution for a system of inequalities. The solution of a variable in an equality will have set of values. The solution of an inequality is expressed in interval notation and some times in set builder notation.

### Solved Example

Question: Solve x2 - x - 6 < 0

Solution:
x2 - x - 6 < 0

Step 1:
Factors of x2 - x - 6

x2 - x - 6 = (x – 3)(x + 2)

Step 2:

(x – 3)(x + 2) < 0

=> (-$\infty$, -2), (-2, 3) and (3, $\infty$) are expected intervals.

Test intervals to determine whether the value satisfies the original inequality. If so, we can conclude that the interval is a solution of the inequality.

The inequality is satisfied for all x values in (-2, 3).

## Solutions of Inequalities

Solving an inequality is the process of finding the numbers that make an algebraic inequality a true numerical statement. Such numbers are called as solutions of the inequality. There are various ways to display the solution set of an inequality. The three most common ways to show the solution set are the set builder notation, a line graph of the solution and interval notation. The graph of inequalities will give a very clear picture about its solution. The graphing is done on a number line in case of single variable. The general process for the solving inequality closely parallels the process for solving equations. Additive property and multiplicative property of inequality helps to find the solutions of the inequalities.

### Solved Example

Question: Solve 5x + 7 > 27 and graph the solutions.

Solution:
Given inequality 5x + 7 > 27

Subtract 7 from both sides of the equation

=> 5x + 7 - 7 > 27  - 7

=> 5x > 20

Divide each side by 5

=> $\frac{5x}{5} > \frac{20}{5}$

=> x > 4

The solution set is (4, $\infty$)

The graph of the solution set: