When faced with an absolute value equation, we need to correctly utilize the definition of absolute value.

Definition:  Let \(x\) be some variable or algebraic expression.  Then

\(\left| x \right| = \left\{ {\begin{array}{*{20}{c}}
  { - x,     if  x < 0} \\
  {x,     if  x \geqslant 0}
\end{array}} \right.\)

We can state this in the form of an equation as

\(\left| x \right| = a \Leftrightarrow x =  - a{\text{  or  }}x = a\)

That is, if \(x\) is a negative number, then the absolute value of \(x\) its opposite.  But if \(x\) is a positive number, then the absolute value of \(x\) is itself.  Unless \(x = 0\), every absolute value equation should have two solutions.  Let’s see if we can solve absolute value equations now.

Example:  Solve the equation \(\left| {10x - 1} \right| = 51\)

Solution:  We know that the expression \(\left| {10x - 1} \right|\) has a different result depending on whether \(10x - 1 0\) or \(10x - 1 \geqslant 0\). And we don’t know which one is true in this situation.  So we set up two equaions.  We have

So our solution set for the equation \(\left| {10x - 1} \right| = 51\) is \(\left\{ { - 5,\Large \frac{{26}}{5}} \right\}\).

This may seem a bit confusing at first, but let’s check the answers and you’ll see that they’re true!  We have

\(\begin{gathered}
  \left| {10\left( { - 5} \right) - 1} \right| = 51  \\
  \left| { - 50 - 1} \right| = 51  \\
  \left| { - 51} \right| = 51  \\
\end{gathered} \)

This solution is certainly true.  Let’s check the other.  We have

\(\begin{gathered}
  \left| {10\left( {\frac{{26}}{5}} \right) - 1} \right| = 51  \\
  \left| {2\left( {26} \right) - 1} \right| = 51  \\
  \left| {52 - 1} \right| = 51  \\
  \left| {51} \right| = 51  \\
\end{gathered} \)

Another true solution!  Hopefully you can see and trust the fact that each absolute value equation has two solutions!  Let’s do one more example:

Example:  Solve the equation \(\left| { - 6 - p} \right| = 4\).

Solution:  This statement implies

So the solution set for the equation \(\left| { - 6 - p} \right| = 4\) is \(\left\{ { - 10, - 2} \right\}\).

Below you can download some free math worksheets and practice.