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.
- Example-1: Solve the inequality,
-5x + 7 ≤ -13
-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
(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.