Thursday, June 9, 2016

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.

  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
Positive absolute value expression:|3x+2|=3x+2
Positive absolute value expression:|x-1|=x-1
Negative (<0) 3x+2<0
Positive absolute value expression:|3x+2|=-(3x+2)
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{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.

For x≥1:

Step 4: Combine the ranges


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,


  1. I review the critique of qualitative research article and can say that if we get help from expertise we can do this our self. With little help things things can be so easy.

  2. Do you know how to write a term paper ? Navigate to this page and you will find a detailed guide!

  3. Your new valuable clues advocate bounty someone like American state and strongly bigger to my work force in geographic point structures. With thanks a decent arrangement a top notch bargain; from all and differed family.
    Veterans Day Quotes
    Veterans Day Clipart
    Happy Veterans Day
    Happy Veterans Day

  4. Our top essay services are focused on making your life in school smooth and enjoyable in comparison to what you might be going through at the moment. Seek services of topessaywriting today!

  5. We are one of the nursing essay writing service company as we submit our papers on time and students will have a satisfactory time to go over the essay trying to counter check for any mistake or error which might need urgent correction. Order a custom assignment help and get the best results.

  6. This content is very good. Thank you for this great website.

    โปรโมชั่นGclub ของทางทีมงานตอนนี้แจกฟรีโบนัส 50%
    เพียงแค่คุณสมัคร สล็อตออนไลน์ กับทางทีมงานของเราเพียงเท่านั้น
    สมัครสล็อตออนไลน์ >>> Goldenslot
    สนใจร่วมสนุกกับ คาสิโนออนไลน์ คลิ๊กได้เลย
    มีทั้งคาสิโนออนไลน์ หวยออนไลน์ ฟุตบอลออนไลน์ สล็อตออนไลน์ และอื่นๆอีกมากมาย