If c is a positive number, then | x | = c is equivalent to x = c or x = – c. Break the equation up into two equivalent equations using the rule: If | x | = c then x = c or x = - c . If the absolute values of two expressions are equal, then either the two expressions are equal, or they are opposites.| x – 6 | = 4 is equivalent to x – 6 = 4 or x – 6 = – 4 Step 2. x – 6 6 = 4 6 x = 10 x – 6 6 = – 4 6 x = 2 Step 3 . | 10 – 6 | = | 4 | = 4 | 2 – 6 | = | – 4 | = 4 The solutions are x = 10 and x = 2 . Break the equation up into two equivalent equations using the rule: If | x | = c then x = c or x = - c . If x and y represent algebraic expressions, | x | = Step 1.Solving absolute value inequalities is a lot like solving absolute value equations, but there are a couple of extra details to keep in mind.
A very basic example would be as follows: if required.
However, these problems are often simplified with a more sophisticated approach like being able to eliminate some of the cases, or graphing the functions.
When solving equations that involve absolute values, there are two cases to consider.
Case 1: The expression inside the absolute value bars is positive.
To get each piece, you must figure out the domain of each piece.
This method is highly beneficial when the question writer asks for the number of solutions instead of the actual solutions.Thus, the solutions are Sometimes absolute value equations have a ridiculous number of cases and it would take too long to go through every single case.Therefore, we can instead graph the absolute value equations using the definition of absolute value as a piecewise function.The absolute value function exists among other contexts as well, including complex numbers.Note that and For complex numbers , the absolute value is defined as , where and are the real and imaginary parts of , respectively.Let's work through some examples to see how this is done. Another benefit of this graphing technique is that you do not need to verify any of the solutions--since we are only graphing the pieces that are actually mathematically possible, we get all the solutions we are looking for, no less and no more.If you could not discern the solutions from the picture, you can simply solve the equation for each case.If you are not a member or are having any other problems, please contact customer support.This equation is asking us to find all numbers, x , that are 3 units from zero on the number line.Because we are multiplying by a positive number, the inequalities will not change: −2 ≤ x ≤ 6 Done!Equations with a variable or variables within absolute value bars are known as absolute value equations .