The process to solve a linear inequality in one variable is to use the following three properties of inequalities to generate a sequence of inequalities each equivalent to the previous inequality until a simplest inequality is obtained.
Properties of Inequalities:
- If any expression is added to both sides of an inequality the resulting inequality is equivalent to the original inequality.
- If both sides of an inequality are multiplied by the same positive real number, the resulting inequality is equivalent to the original inequality.
- If both sides of an inequality are multiplied by the same negative real number and the inequality symbol is reversed, the resulting inequality is equivalent to the original inequality.
If all of the above observations (Facts A, B, C, D, and E) are put together we arrive at the following single easy process for solving any one (and in fact all) of |M| < k, |M| = k, or |M| > k where M is some linear expression and k is a positive real number
- Step 1: Regardless of the question being asked, focus on the “less–than” inequality.
- Step 2: Convert that inequality to the equivalent compound inequality.
- Step 3: Solve the compound inequality.
- Step 4: Write the solution set and graph it (at least visualize it) on the Real Number Line.
- Step 5: Identify the endpoints of this solution set as the elements of the solution set for the corresponding boundary equation.
- Step 6: Identify the remainder of the Real Number line as the solution set for the “greater–than” inequality.
It is worth observing again that this approach always explicitly produces the solutions set to all three possibilities (<, =, >).