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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