How to Solve Absolute Value Inequalities

Don't let absolute value symbols in an inequality have you throwing your hands up in the air in frustration. This article will guide you through how to solve absolute value inequalities without ripping your hair out.

Things You'll Need

• Paper
• Pencil

Instructions

Solving Absolute Value Inequalities When the Equation is Greater Than

1. Look at the absolute value inequality that you are given, and make note of the processes that are in the equation.Is there subtraction or addition? Are you going to have to multiply or divide? Are there any exponents? Knowing what processes are in the absolute value inequality will let you know what processes you will need to do and it will also let you begin to decide what processes will need to be done before others. Remember that when you are solving any equation for a variable you will need to do the reverse order of operations to both sides of the equation in order to get your solution. This means that you will begin with addition/subtraction, then move to multiplication/division, followed by exponents, and then finish up with any parentheses. An easy way to remember this is the normal order of operations is PEMDAS, so the reverse order is SADMEP, with each letter standing for a process.

2. Acknowledge what the absolute value symbols mean for the inequality that you are about to solve. When you are dealing with a "great than" inequality such as | 3x –7 | > 10 you have to not only solve the inequality 3x-7>10, you also have to solve the inequality 3x-7<-10 because the absolute value symbols will make any negative solutions positive.

3. Begin to solve the two inequalities using the reverse order of operations.We have the two equations that we came up with in the previous step, 3x-7>10 and 3x-7<-10.Now we will use the reverse order of operations on each inequality, we begin by adding seven to both sides of both inequalities which leaves us with the following two inequalities:3x>17 and 3x<-3

4. Continue using the reverse order of operations to solve the two inequalities to get your solutions to the original absolute value inequality.Divide both sides of both inequalities by 3, which will leave you with the following solutions:x>17/3 and x<-1.

5. Remember, if you have to divide or multiply by a negative number you need to flip the inequality symbol.

Solving Absolute Value Inequalities When the Equation is Less Than

1. Follow Step 1 from above.

2. Acknowledge what the absolute value symbols mean for the inequality.If we were solving the problem from above, only the problem was a "less than" equation we would need to set up the inequality as follows to account for the absolute value symbols,We would begin with the absolute value inequality, | 3x –7 | < 10, which would then become -10<3x-7<10 because the absolute value symbols turn negative solutions into positive solutions.

3. Use the reverse order of operations like we did with the "greater than" equation to solve for the solutions.Add 7 to all three parts of the equation:-3<3x<17Divide all three parts of the equation by 3 to get our solution:-1<x<17/3.

4. Remember, if you have to multiply or divide by a negative number to get your solution you also need to flip the inequality symbols.