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Crystal Wheeler 2022-07-09 Answered
How do you solve: x 2 - 9 x 2 - 1 < 0 ?
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Answers (1)

fugprurgeil
Answered 2022-07-10 Author has 12 answers
Step 1
Your inequality looks like this
x 2 - 9 x 2 - 1 < 0
Right from the start, you know that any solution set that you might come up with cannot include the values of x that will make the denominator equal to zero.
More specifically, you need to have
x 2 - 1 0 x ± 1
Now, in order for this inequality to be true, you need to have
x 2 - 9 < 0 and x 2 - 1 > 0
or
x 2 - 9 > 0 and x 2 - 1 < 0
For the fist set of conditions to be true, you need to have
{ 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 ) .
For the second set of conditions, you need to have
{ 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 values of x that belong to this interval will make the numerator negative and the denominator positive, which in turn will make the fraction negative.
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