Section 1.3 Absolute Value Equations and Inequalities

Section Objectives

  1. Solve absolute value equations.
  2. Solve absolute value inequalities.



Absolute Value

The absolute value of a number, , is that number's distance from zero on the number line. The absolute value of is written , and this quantity is always either positive or zero. In fact,



Based on the definition of absolute value, it is clear that or , depending on whichever is positive. If we're not sure which is positive, we must consider both options.


For example...

If , then must either be or .




Absolute Value Equations

Suppose is positive or zero. The absolute value equation is simply a compound equation in disguise:



To solve an absolute value equation:

  1. Use algebra to write the equation in the form . In this context, and may be entire algebraic expressions, but the equation can only make sense if .
  2. Solve the compound equation .
  3. Check your answers.


Examples

Solve for :




Solve for :




Solve for :




Solve for :




Solve for :




Solve for :




Absolute Value Inequalities

Just as we did with absolute value equations, we'll solve absolute value inequalities by rewriting them as compound inequalities. There are two cases to consider (assuming ):

These ideas should make sense if you think about representing a distance from zero.



Important idea: Be on the lookout for equations and inequalities that are never true or always true. Usually you can spot these before you even start the solution process.



Examples

Solve for :




Solve for :




Solve for :




Solve for :




Solve for :