Solving Absolute Value Equations and Inequalities

This section covers:

As we saw earlier, an absolute value (designated by |  |) means take the positive value of whatever is between the two bars.  The absolute value is always positive, so you can think of it as the distance from 0.  So  and .  It’s as simple as that!

(Note that we also address absolute values here in the Solving Algebraic Equations section,  here in the Linear Inequalities section, here in the Piecewise Functions section, and  here in the Graphing Rational Functions,  including Asymptotes section.)

Solving Absolute Value Equations

Solving absolute equations isn’t too difficult; we just have to separate the equation into two different equations (once we isolate the absolute value), since we don’t if what’s inside the absolute value is positive or negative (we can do this with a number line if we want).

I like to then make the expression on the right hand side (without the variables) both positive and negative and split the equation that way.

The other thing we have to remember is that we must check our answers, since we may get extraneous solutions (solutions that don’t work).

There are a few cases with absolute value equations or inequalities that you may see where you don’t have to go any further.  One is when we have isolated the absolute value, and it is set equal to a negative number, such as  \(\left| {x-5} \right|=-4\),  or  \(\left| {x-5} \right|\le -4\),  for example.   Since an absolute value can never be negative, we have no solution for this case.

The other is when the absolute value is greater than a negative number, such as  \(\left| {x-5} \right|>-4\)  for example.  In this case our answer is all real numbers, since an absolute value is always positive.

Here are more problems:

Solving Equations with Absolute Values

Here’s one more that’s a bit tricky, since we have two expressions with absolute value in it.  In this case, we have to separate in four cases, just to be sure we cover all the possibilities.  We then must check for extraneous solutions, possible solutions that don’t work.

Solving with Absolute Value

Here’s another way to approach the absolute value problem above, using number lines (sort of like sign charts):

Solving Absolute Values using Number LinesSolving Absolute Value Inequalities

When dealing with absolute values and inequalities (just like with absolute value equations), we have to separate the equation into two different ones, if there are any variables inside the absolute value bars.

We first have to get the absolute value all by itself on the left.

Now we have to separate the equations.  We get the first equation by just taking away the absolute value sign away on the left.  The easiest way to get the second equation is to take the absolute value sign away on the left, and do two things on the right:  reverse the inequality sign, and change the sign of everything on the right (even if we have variables over there).

We also have to think about whether or not to use “or” or “and” between the two new equations.  The way I remember this is that with a > or  sign, you can put a little “o” in it after the absolute value, so you want to use the “or” (or remember, greater than starts with “g”, so think “gore”).  For < or  signs, use “and” (you can remember less than starts with “l”, so think “land”).

Or or And

And again, if we get something like  \(\left| {x+3} \right|<0\),  there is no solution,  and something like  \(\left| {x+3} \right|\ge -4\),  there are infinite solutions (all real numbers).

Here are some examples:

Absolute Value Inequalities

Here are more examples that have absolute value rational function inequalities:

Absolute Value Inequalities with Rational Functions

Rational Inequality with Absolute Value and Quadratic Formula

Graphs of Absolute Value Functions

Note that you can put absolute values in your Graphing Calculator (and even graph them!) by hitting MATH, scroll right to NUM, and then hitting 1 (abs) or ENTER.

Absolute Value functions typically look like a V (upside down if the absolute value is negative), where the point at the V is called the vertex.  For the absolute value parent function, the vertex is at (0, 0).

We looked at absolute value parent functions and their transformations here in the Parent Functions and Transformations section, and absolute value functions as piecewise equations here in the Piecewise Functions section.

Note that the general form for the absolute value function is  \(f\left( x \right)=a\left| {x-h} \right|+k\),  where (h, k) is the vertex.  If a is positive, the function points down (like a V); if a is negative, the function points up (like an upside down V).

Here’s a graph of the parent function, and also a transformation:

Absolute Value Graphs

Applications of Absolute Value Functions

Absolute Value Functions are in many applications, especially in those involving V-shaped paths and margin of errors, or tolerances.

Here’s an example of a basic absolute value inequality “word” problem that you may see:

Simple Absolute Value Word Problem

Here are more examples that are absolute value applications:

Absolute Value Word Problems Learn these rules, and practice, practice, practice!

Click on Submit (the arrow to the right of the problem) to solve this problem. You can also type in more problems, or click on the 3 dots in the upper right hand corner to drill down for example problems.

If you click on “Tap to view steps”, you will go to the Mathway site, where you can register for the full version (steps included) of the software.  You can even get math worksheets.

You can also go to the Mathway site here, where you can register, or just use the software for free without the detailed solutions.  There is even a Mathway App for your mobile device.  Enjoy!

On to Solving Radical Equations and Inequalities – you’re ready!