How To Solve Absolute Value Equations and Absolute Value Inequalities

Solving absolute value equations and absolute value inequalities is not as hard as it sounds. There are several steps to solving most equations or inequality involving absolute value. The easiest of absolute value equations is where the absolute value expression is isolated on one side of the equation and is set equal to a negative number. Because the absolute value is measuring distance, it must be positive and thus cannot be equal to a negative number.

Example 1 Solving an Absolute Value Equation without a Solution

A symbolic representation of an absolute value equation set equal to a negative number. At the outset of this example, the absolute value expression is not isolated and it requires two steps to solve for the absolute value expression.

As I wrote earlier, this is the easiest of examples. There is only one other type of absolute value equation and that where there is a solution. You can only recognize there is a solution to an absolute value equation when the absolute value expression is by itself on one side of the equation. Normally, the absolute value expression is kept on the left hand side. See example 2 below. Once it is determined that there is a solution (1), it must be written as a compound equation using the word ‘or’. ‘Or’ is a funny little word and has its roots in logic and that is how it fits into algebra. All positive real number will give two solutions to an absolute value equation. With that being said, once the absolute value expression has been isolated it is time to write the compound equation (2) and (3). The first equation is the same as the original equation without the absolute value symbols (2). The second equation is the same except for the lone constant term on the right side. It needs to be the opposite (3). To solve the compound equation you need a strong background in algebra 1, add 7 to both sides of both equations (4) and divide each side by of each equation by -4 (5). The solution set is -3 and -0.5 (6).

Example 2 Solving and Absolute Value Equation with a Solution


That is all for tonight. I am going to give my algebra two students a choice on their review assignment tomorrow. They will be able to choose to do a whole mess of problems or pick one problem from each section of the review and write a blog post like I did here. They will not have to make the cool images, but it would be nice. I will let you know how it goes.

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