GRE Algebra | Solving Linear Inequalities

Last Updated : 25 Apr, 2019

A linear inequality is an inequality which involves a linear function and contains the following symbols:

< less than
> greater than
≤ less than or equal to
≥ greater than or equal to

A linear inequality is same as a linear equation, except the equals sign of equation replaced with an inequality symbol. For example, 2x - 2 ≤ 9, is a linear inequality in one variable, which states that "2x - 2" is "less than or equal to 9".

Solution Set is the set of values of an inequality that make its value true.
Equivalent inequalities are the inequalities having same solution set.

The rules to solve linear inequality are:

When same constant added to or subtracted from both sides of an inequality, direction preserved and the new equality is equivalent to the original.
When an inequality is multiplied or divided by the same non-zero positive constant on both sides, the direction of the inequality is preserved but if constant is negative then the direction is reversed.

Examples:

Example-1: Solve the inequality,
```
-5x + 7 ≤ -13 
```
Solution:
```
-5x + 7 ≤ -13
-5x ≤ -20 
```
Multiply both sides by (-1) then inequality symbol changes, so,
```
5x ≥ 20

Hence,
x ≥ 4 
```
Therefore, the solution set of -5x + 7 ≤ -13 consists of all the real numbers greater than or equal to 4.
Example-2: Solve the inequality,
```
(2y + 9)/7 > 11 
```
Solution:
```
(2y + 9)/7 > 11 
2y + 9 > 77
2y > 68
y > 34 
```
Therefore, the solution set of (2y + 9)/7 > 11 consists of all the real numbers greater than 34.