Solving equations and inequalities with absolute value Lesson 17

Absolute value The absolute value of a number is the distance along the x-axis from the origin to the graph of the number. The absolute value of every number except zero is greater than zero.

Solving absolute value equations Use inverse operations to isolate the absolute value expression on one side of the equation. Set the expression inside the absolute value symbol to equal the other side and also to equal the opposite of the other side. Ex: if l x-a l = k then x-a = k or x-a = -k

Doing this may result in extraneous solutions ( solutions that do not satisfy the original absolute value equation). You must check all possible solutions

Solving absolute value equations Solve l x-6 l = 4 and graph the solution x-6 = 4 or x-6 = -4 x= 10 or x= 2 Check l10-6l = 4 yes l 2-6 l = 4 yes Graph the 2 points on the number line

example Solve l 3x+1 l -4 = 6 Solve l 9x+7 l = -2 Solve l 4x+12 l = 8x Solve l 3x+18 l = -10

Solving absolute value inequalities with conjunctions Absolute value inequalities are either conjunctions or disjunctions. Conjunctions are absolute value statements with < Disjunctions are absolute value statements with >

Solve and graph l x-5 l -2 x 3 Graph l 2x-5 l >9 so 2x-5>9 or 2x-5<-9 2x >14 2x< -4 x>7 or x<-2 graph

practice Solve l 3x-6 l >3 Solve - l 5x-4 l <6 Solve - l4x-2 l <1

Parent functions A parent function is the simplest function of a particular type. New functions can be graphed by transformations, or changes, to the graph of the parent function. These can involve changes in size, shape, orientation, or position of the parent function

Transforming f(x) = l x l Graph F(x) = l x l F(x) = - l x l F(x) = 1/2 l x l F(x) = l x-4 l F(x) = l x+4 l F(x) = l x l - 4 F(x) = l x l + 4