Solving Absolute Value Equations Unit 1A Lesson 4

The Absolute Value Function is a famous Piecewise Function. It has two pieces: below zero: – x from 0 onwards: x f(x) =|x| f(x) = |x| = x, if x > 0 – x, if x < 0

EXAMPLE 1 Solve | x + 2 | = 7 (x + 2) = 7 –(x + 2) = 7 x + 2 = 7 –x – 2 = 7 x = 5 –9 = x x = –9 (5, 7)(– 9, 7)

EXAMPLE 2 (– 2,3) (5, 3)

EXAMPLE 3

EXAMPLE 4 We can’t get a negative value out of the absolute value. Since this isn’t possible that means there is no solution to this equation.

Practice

To this point we’ve only looked at equations that involve an absolute value being equal to a number, but there is no reason to think that there has to only be a number on the other side of the equal sign. If both sides of the equation contains a variable a CHECK must be done to rule out extraneous roots.

CHECK STEP 1

CHECK STEP 2

CHECK STEP 1

CHECK STEP 2

Practice There is NO solution

Likewise, there is no reason to think that we can only have one absolute value in the problem. So, we need to take a look at a couple of these kinds of equations. Again a CHECK must be done to rule out extraneous roots.

STEP 1: Both inside values are EQUAL CHECK

STEP 2: Both inside values are EQUAL but with OPPOSITE signs Since both sides gave the same result you only have to do ONE SIDE!!!

CHECK