# GRE Algebra | Solving Linear Inequalities

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.

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