SOLUTION: Hi. I'm really struggling on the the process required to solve the following absolute value inequality: absolute value of x squared minus 2 is less than 1 I want to not only

Algebra ->  Absolute-value -> SOLUTION: Hi. I'm really struggling on the the process required to solve the following absolute value inequality: absolute value of x squared minus 2 is less than 1 I want to not only       Log On


   

Question 216323: Hi. I'm really struggling on the the process required to solve the following absolute value inequality:
absolute value of x squared minus 2 is less than 1
I want to not only know the answer but exactly what it is you do to get to the answer and why in depth. Sorry for not specifying earlier. Thanks.
Answer by drj(1380) About Me  (Show Source):
You can put this solution on YOUR website!
Hi. I'm really struggling on the the process required to solve the following absolute value inequality:
absolute value of x squared minus 2 is less than 1

abs%28x%5E2-2%29%3C1

Step 1. Yep, absolute value is a tricky concept. Think of absolute value as a distance measurement. Say you walk five steps forward the distance is 5 steps. Say you walk five steps backward the distance is still 5 steps. That is abs%28-5%29=5.
The absolute value will lead to two inequalities.

Also, when taking the x%5E2 will always be positive even if x is negative. So when you take the sqrt%28x%5E2%29=abs%28x%29. Please try to understand this concept.

Step 2. The absolute value leads to two equations given as follows

x%5E2-2%3C1 Equation A where x%5E2-2 is positive

2-x%5E2%3C1 Equation B where x%5E2-2 is negative. To make it positive we reverse the positions in x%5E2-2 to 2-x%5E2.

Step 3. Let's start with Equation A

Add 2 to both sides of inequality

x%5E2-2%2B2%3C1%2B2

x%5E2%3C3

Taking the square root to both sides yields

abs%28x%29%3Csqrt%283%29

which also leads to two more equations as

x%3Csqrt%283%29 for positive x and

-x%3Csqrt%283%29 for negative x or
x%3E-sqrt%283%29 where we mulitplied -1 to both sides of the equation and reversed the inequality.

We can combine the two inequalities as

ANSWER: -sqrt%283%29%3Cx%3Csqrt%283%29

Step 4. For equation B, let's add x%5E2-1 to both sides of the inequality

2-x%5E2%2Bx%5E2-1%3C1%2Bx%5E2-1

1%3Cx%5E2

Take the square root to both sides of the equation.

sqrt%281%29%3Cabs%28x%29 or 1%3Cabs%28x%29

This also leads to two more equations

1%3Cx where x is for positive number and

1%3C-x where x is for negative numbers or
-1%3Ex where we multiplied by -1 and reversed the inequality.

ANSWER: 1%3Cx and -1%3Ex

I hope the above steps were helpful.

For FREE Step-By-Step videos in Introduction to Algebra, please visit
http://www.FreedomUniversity.TV/courses/IntroAlgebra and for Trigonometry visit
http://www.FreedomUniversity.TV/courses/Trigonometry.

Good luck in your studies!

Respectfully,
Dr J