Absolute value equations

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

 $$10x - 1 = - 51$$ or $$10x - 1 = 51$$ $$10x = - 50$$ or $$10x = 52$$ $$x = - 5$$ or $$x = \Large \frac{{52}}{{10}} = \Large \frac{{26}}{5}$$

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

 $$- 6 - p = - 4$$ or $$- 6 - p = 4$$ $$- p = 2$$ or $$- p = 10$$ $$p = - 2$$ or $$p = - 10$$

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

5308 x

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3675 x

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4820 x

Solve each equation.

This free worksheet contains 10 assignments each with 24 questions with answers.

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