Table of Contents

## Equations

In Section 1.5, we learned that

\[ \bbox[#F2F2F2,5px,border:2px solid black]{\large |x|=c\geq 0\qquad \Leftrightarrow \qquad x=\pm c}\]

provided $c\geq0$. So to solve equations involving an absolute value follow these steps:

- Isolate the absolute value expression on one side and the rest of terms on the other side. That is, rewrite the equation as

\[

|P|=Q

\] where $P$ and $Q$ are two expressions in $x$ [to indicate the dependence on $x$, we may write them as $P(x)$ and $Q(x)$]. - Equate the expression inside the absolute value notation once with + the quantity on the other side and once with – the quantity on the other side.

\[

P=Q\qquad\text{or}\qquad P=-Q

\] - Solve both equations.
- Check your answers by substitution in the original equation.

- When $Q<0$, the equation will not have a solution because always $|P|\geq0$. When $Q$ is an expression, we need to substitute the solutions in $Q$ to make sure that $Q\geq0$.

When there are more than one absolute value, for example when we have

\[

|P|+|R|=Q

\]
where $P, Q$, and $R$ are some expressions, the above technique may not work. In such cases, we need to find where $P$ and $R$ are positive and where they are negative and then solve the equation in the same way that we solve regular equations.

## Inequalities

To solve absolute value inequalities, recall (see Section 1.4):

- $|x|<c$ is equivalent to $-c<x<c$.
- $|x|>c$ is equivalent $x>c$ or $x<-c$

where $c$ is a positive number.

The above equivalent statements hold true if we replace $<$ by $\le$ and $>$ by $\ge$.