# 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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