How do you solve: <mstyle displaystyle="true"> x 2

Crystal Wheeler

Crystal Wheeler

Answered question

2022-07-09

How do you solve: x 2 - 9 x 2 - 1 < 0 ?

Answer & Explanation

fugprurgeil

fugprurgeil

Beginner2022-07-10Added 12 answers

Step 1
Your inequality looks like this
x 2 - 9 x 2 - 1 < 0 
You know right away that the values of x that will cause the denominator to equal zero cannot be part of any solution set you might come up with.
More specifically, you need to have
x 2 - 1 ≠ 0 ⇒ x ≠ ± 1 
This disparity must exist in order for it to be true.
x 2 - 9 < 0    and    x 2 - 1 > 0 
or
x 2 - 9 > 0    and    x 2 - 1 < 0 
The first set of requirements must be satisfied in order for
{ x 2 - 9 < 0 ⇒ x < ± 3 ⇒ x ∈ ( - 3 , 3 ) x 2 - 1 > 0 ⇒ x > ± 1 ⇒ x ∈ ( - ∞ , - 1 ) ∪ ( 1 , + ∞ ) 
This means that you need x ∈ ( - 3 , - 1 ) ∪ ( 1 , 3 ) .
You must meet the second set of requirements in order to
{ x 2 - 9 > 0 ⇒ x > ± 3 ⇒ x ∈ ( - ∞ , - 3 ) ∪ ( 3 , + ∞ ) x 2 - 1 < 0 ⇒ x < ± 1 ⇒ x ∈ ( - 1 , 1 ) 
This time, those two intervals will not produce a valid solution set, or x ∈ ∅ 
The only option left to you is x ∈ ( - 3 , - 1 ) ∪ ( 1 , 3 ) . The fraction will be negative because the values of x that fall within this interval will result in a positive denominator and a negative numerator.

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