Solving equations and inequalities with absolute value Lesson

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 l 10 -6 l = 4 yes l 2 -6 l = 4 yes Graph the 2 points on the number line

example • • Solve l 3 x+1 l -4 = 6 l 9 x+7 l = -2 l 4 x+12 l = 8 x l 3 x+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 so x-5< 2 and x-5> -2 • x < 7 and x >3 • Graph • l 2 x-5 l >9 so 2 x-5>9 or 2 x-5<-9 • 2 x >14 2 x< -4 • x>7 or x<-2 • graph

practice • Solve l 3 x-6 l >3 • Solve - l 5 x-4 l <6 • Solve - l 4 x-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) = F(x) = lxl -lxl 1/2 l x l l x-4 l l x+4 l lxl-4 lxl+4