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.