With this concept in mind, we can begin to discuss how having an absolute value equation or inequality compares to usual linear equations and linear inequalities.
Solving Absolute Value Equations vs Solving Linear Equations
If we have an absolute value equation we can have two solutions. For example, in the equation |x|=2 x can be 2 or -2, since both are 2 units away from 0 on the number line.
To take both possibilities into account when solving an absolute value equation, we isolate the expression inside the absolute value and then derive 2 equations — one using the negative value, and the other using the positive value. We can see an example below.
Now, we can derive the positive and the negative case.
Derived Eqautions 0.5cm [0.8em]
x+4 = 5x 1cm x+4 =-5x
If we are solving an absolute value equation and one of the derived equations solutions is not a solution to the original equation, it is called an extraneous solution. We can summarize what we mention above in the following table.
Solving Absolute Value Equations
Solving Linear Equations
We use Properties of Equality to isolate the absolute value expression, then we derive two equations.
We use Properties of Equality to isolate the variable, then we get our solution.
It can have 2 solutions; one for each of the derived equations, no solutions, or extraneous solutions.
It can have 1 or no solution.
Solving Absolute Value Inequalities vs Solving Linear Inequalities
