High School Math Solutions – Inequalities Calculator, Absolute Value Inequalities Part II

Last post we talked about absolute value inequalities with one absolute value expression. In this post, we will learn how to solve absolute value inequalities with two absolute value expressions. To do this, we will use some concepts from a previous post on rational functions. This can get a little tricky and confusing, so take time to read everything carefully.

I’ll show you the steps to solving these inequalities and then I will go through one example step by step.

Steps:

1. For each absolute value expression, figure out their negative and positive ranges.
2. Combine ranges if needed
3. For each range, evaluate the absolute value inequality
4. Combine the ranges

Here’s our first example (click here):

\frac{|3x+2|}{|x-1|} >2

Step 1: For each absolute value expression, figure out their negative and positive ranges

	3x+2	x-1
Positive (≥0)	3x+2≥0 Range:x\ge-\frac{2}{3} Positive absolute value expression:|3x+2|=3x+2	x-1≥0 Range:x≥1 Positive absolute value expression:|x-1|=x-1
Negative (<0)	3x+2<0 Range:x\le-\frac{2}{3} Positive absolute value expression:|3x+2|=-(3x+2)	x-1<0 Range:x<1 Positive absolute value expression:|x-1|=-(x-1)

Step 2: Combine ranges if needed

Let’s see our ranges: x≥-\frac{2}{3},\quad x≥1,\quad x<-\frac{2}{3},\quad x<1
Now let’s combine any ranges if possible: x<-\frac{2}{3},\quad -\frac{2}{3}≤x<1, \quad x≥1

Step 3: For each range, evaluate the absolute value inequality

For x<-\frac{2}{3}:

\frac{-(3x+2)}{-(x-1)}>2	We use the negative absolute value expressions because for both expressions the range is less than their negative range.
\frac{3x+2}{x-1}>2
\frac{x+4}{x-1}>0	Now, we see that this is just a rational inequality. So we solve the rational inequality
x<-4\:or\:x>1	The steps aren’t shown to get the answer. You can refer back to the rational inequalities blog if you forget how to solve it

For -\frac{2}{3}≤x<1:

\frac{3x+2}{-(x-1)}>2	We use the negative absolute value expression for the |x-1| because the range is less than its range.
\frac{5x}{1-x}>0
0<x<1

For x≥1:

\frac{3x+2}{x-1}>2
\frac{x+4}{x-1}>2
x<-4\:or\:x>1

Step 4: Combine the ranges

(x<-\frac{2}{3}\:and\:x<-4\:or\:x>1)\:or\:(-\frac{2}{3}≤x<1\:and\:0<x<1)\:or\:(x≥1\:and\:x<-4\:or\:x>1)

Answer: x<-4 or x>1 or 0<x<1


Hopefully, that wasn’t too complicated. Now let’s see one more example.

Last example (click here):

As you can see, this topic can be very confusing and tricky. The more you practice, the more natural it will become. For more practice examples check out Symbolab’s practice.

Until next time,
Leah