3.8 Chapter Summary

In this chapter, we studied the concept of distance from a mathematical perspective.

Page Summary
Introduction The distance between two objects is measured using a mathematical symbol known as the absolute value.
Absolute Value Definition The absolute value function, |x|, is a piecewise defined function; it is defined differently on different parts of its domain:

f(x)=\begin{Bmatrix} x & x\geq 0\\ -x& x<0 \end{Bmatrix}

Graphically, it looks like:

Distance from the Origin If a≥0, then to solve the equation |x|=a, we can find x by either

1. Moving left or right "a" units on the number line.
2. Solving the two equations x=a or x=-a

If a<0, then the equation |x|=a has no solution.

Solving Absolute Value Equalities To solve more complicated absolute value equalities, we can follow the basic process that we previously studied. However, since the expression in the absolute value could be very complicated, in these cases, it's better to find the solution algebraically as opposed to using the number line method.
Distance Between Two Points In English, |a-b| represents the distance between the points a and b.
Solving Less Than Inequalities

|x| < a means that x is less than "a" units from the origin.
If a>0, we can solve the absolute value inequality |x|< a, by solving the compound statement:

x>-a {\rm\; and\;}x<a
Sovling Greater Than Inequalities

|x|> a means that x is more than "a" units from the origin.
If a>0, we can solve the absolute value inequality |x|> a, by solving the compound statement:

x<-a{\rm \; or \;}x> a

Key Points