How to Solve Combined Inequalities

Solving combined inequalities is just like solving normal inequalities, but with two separate inequalities combined into one. The main concept to know is the difference between the words "AND" and "OR," which control or limit the answers. "AND" means the solution will be an intersection where both inequalities hold true. "OR" means the solution will be an intersection where only one OR the other inequality holds true. An example follows each step in parentheses.

Difficulty: Moderately Easy

Instructions

Combined Inequalities Using "AND"

1. Write down the combined inequality on paper. (-6 < 2x + 2 < 8)

2. Separate the combined inequality into its individual inequalities. (-6 < 2x + 2 AND 2x + 2 < 8)

3. Solve each inequality for the variable. (x > -4 AND x < 3)

4. Write the answer in combined inequality form. (-4 < x < 3, all answers are at the intersection where the answers cross, so that both rules are fulfilled.)

5. Graph the answers on a number line to show exactly what the answers mean. (We use open dots at -4 and 3 with a line between them since all answers lie BETWEEN -4 and 3.)

Combined Inequalities Using "OR"

6. Write down the combined inequality on paper. (-6 > 2x + 2 OR 2x + 2 > 8)

7. Separate the combined inequality into its individual inequalities. (-6 > 2x + 2 OR 2x + 2 > 8)

8. Solve each inequality for the variable. (x < -4 OR x > 3)

9. Write the answer in combined inequality form. (x < -4 OR x > 3, all answers fulfill one OR the other rule is fulfilled.)

10. Graph the answers on a number line to show exactly what the answers mean. (We use open dots at -4 and 3 with a line and arrow pointing outward to infinity since no answers cross between -4 and 3.)

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