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Solving Linear Inequalities

An inequality is a statement indicating that two expressions are not equal to one another in a particular way (e.g. one expression is larger or smaller than the other). In the case of a linear inequality, it can be simplified into the form

ax ≥ b, ax b, ax ≤ b, or ax b

where a and b are real numbers and a ≠ 0. 

Solving a Linear Inequality

To solve a linear inequality, you have to isolate for the variable by doing the following steps:

  • Expand (if applicable)
  • Group like terms (if applicable)
  • Rearrange so that all terms with the variable in them are on one side of the inequality while all the terms without the variable in them (i.e. just number terms) are on the other side. That is, rearrange into the form ax > b or ax b, etc. (to do this, you're just adding/subtracting terms from both sides)
  • Divide by the coefficient of the variable to solve for the variable (i.e., once you've got ax > b or ax b, etc., divide both sides by a to solve for x). IMPORTANT: if you divide by a negative number, the inequality switches direction (i.e., > becomes < and so on)

Example: Solve the linear inequality 5x - 1  3x + 2


5x - 1  3x + 2

5x - 3x  2+ 1

        2x  3

         x  3/2

Example: Solve the inequality -6x + 1 < 3x + 4


-6x +1 < 3x + 4

-6x - 3x < 4 - 1

       -9x < 3

          x > -1/3

Note that we could have avoided the situation having to divide by a negative number (and hence changing the sign of the inequality) by collecting the terms with the variable on the right of the inequality and the terms with only numbers on the left. Let's do the question this way as well so that you can see the difference:

-6x +1 < 3x + 4

    1 - 4 < 3x + 6x

        -3 < 9x

     -3/9 x

     -1/3 x

Of course, even though it might not look like it at first glance, -1/3 < x and x -1/3 are saying the exact same thing. What we're saying in both cases is: x is greater than - 1/3

Linear Example 1:

Linear Example 2:

Quadratic Example:

Three Factors Example:

Absolute Value Inequalities Example 1:

Absolute Value Inequalities Example 2: