Absolute Value Equations
First let us review the definition of absolute value $|\cdot |$.
Definition. The absolute value of $|a|$ of a number $a$ is defined by
$$|a|=\left\{\begin{array}{ccc}
a & \mbox{if} & a\geq 0\\
-a & \mbox{if} & a<0
\end{array}\right.$$
Interestingly a lot of students get confused with the definition while they have no problem with getting the absolute of a particular number correctly like $|2|=2$ and $|-2|=2$. That’s because of the way they were taught. Many teachers teach absolute value like some sort of magic: “Hey guys what absolute value does is whatever number you put into between the two vertical lines it becomes positive.” While this may be easy to understand for students this is not an action performed in mathematics. In algebra whatever action you do should be carried out by operations such as $+$, $-$, $\times$, or $\div$. In terms of an operation what happens to $|-2|=2$ is in fact that $|-2|=-(-2)=2$. It is not abracadabra. For a letter $a$ representing a number, if $a<0$, to make it positive $|a|=-a$. Students get confused because $-a$ looks like negative but it is not. Remember the condition $a<0$. So $-a$ is actually positive. So don’t be deceived by its look.
Now we are ready to discuss absolute value equations. All you need to know is
\begin{equation}
\label{eq:abs}
|x|=k\ \mbox{if and only if}\ x=\pm k
\end{equation}
for $k>0$. If $k=0$, $x=0$. If $k<0$, obviously there is no solution.
Example. Solve the equation
$$|2x-5|=3$$
Solution. By \eqref{eq:abs}, we get the two linear equations $2x-5=\pm 3$. Solving these equation, we find $x=4$ or $x=1$.
Absolute value equations are related to quadratic equations. If you square the above equation, we get the quadratic equation
$$x^2-5x+4=0$$
Solving this quadratic equation, we of course obtain the same solutions. In practice why bother? Solving absolute value equations directly is easier. Note though that absolute value equations are actually obtained when you solve quadratic equations by the method of completing the square.
Absolute Value Inequalities
For absolute value inequalities all you have to know is the following picture.
Figure 1. Absolute Value Inequalities
From Figure 1 we can read for $k>0$
- $|x|<k$ if and only if $-k<x<k$.
- $|x|\leq k$ if and only if $-k\leq x\leq k$.
- $|x|>k$ if and only if $x<-k$ or $x>k$.
- $|x|\geq k$ if and only if $|x|\leq -k$ or $|x|\geq k$.
Example. Solve the inequality $|x-5|<2$.
Solution. $|x-5|<2$ implies that $-2<x-5<2$ i.e. $3<x<7$.
Example. Solve the inequality $|3x+2|\geq 4$.
Solution. $|3x+2|\geq 4$ implies that $3x+2\leq -4$ or $3x+2\geq 4$ i.e. $x\leq-2$ or $x\geq\frac{2}{3}$.