The graph of an absolute value function is a v-shape.  Why is that?  Consider the basic absolute value function y = |x|.  For positive x or 0, the absolute value of x is just x itself (y = x for x > 0).  For negative x, the absolute value of x is the opposite of x (y = -x for x < 0).  Windows #1-2 show you how you can type in these pieces of function and the resulting graph. 

Instead of representing every absolute value function in this way, we will use the built-in function abs(x). 

What is the solution of the equation |x - 3| = 2 ?  From earlier graphing instruction, we know that the solution is the x-value of the point of intersection.  See the graph of the equation on the left. 

How many points of intersection might there be for any absolute value? 

Looking at the graph on the left, what is the solution of the inequality |x - 3| < 2 ?  We want all x-values where the v-shape is below the line.  The answer is (1,5), that is, all values between but not including 1 and 5. 

 

<Graphing Calculator Explorations
Use algebraic methods to solve |2x + 5| = 3.  Verify your answer using the graphing method described above. Use the graphing method to solve |5x - 4| > 2. Can an expression, that is always positive or zero, ever be less than a negative number?  Use this concept to solve |2x - 1| < -2.  Verify your result by both the algebraic and graphing methods.