Solving Equations and Inequalities with Absolute Value (Exploration: Transforming f(x) = |x|)
Solve absolute value equations and inequalities by splitting into two cases: |ax+b|=c becomes ax+b=c or ax+b=-c, and explore transformations of f(x)=|x| graphically.
Key Concepts
New Concept The absolute value of a number is the distance along the x axis from the origin to the graph of the number.
What’s next Next, you'll use this definition to solve equations and inequalities by breaking them into two distinct cases.
Common Questions
How do you solve an absolute value equation like |3x-2|=10?
Split into two equations: 3x-2=10 and 3x-2=-10. Solve each: 3x=12 gives x=4, and 3x=-8 gives x=-8/3. Both must be checked in the original equation. If either solution makes the expression inside valid, it is a solution.
How do you solve absolute value inequalities like |x-4|<3?
For |expression|<k (with k>0), write the compound inequality -k<expression<k. So |x-4|<3 becomes -3<x-4<3, which gives 1<x<7. For |expression|>k, write two separate inequalities: expression>k or expression<-k.
How do transformations affect the graph of f(x)=|x|?
Vertical shifts (f(x)=|x|+k) move the V-shape up or down. Horizontal shifts (f(x)=|x-h|) move it left or right. Vertical stretches (f(x)=a|x|) widen or narrow the V. Reflection (f(x)=-|x|) flips the V upside down.