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

Santino Bautista 2022-06-27 Answered
How do you solve x - 2 x + 3 < x + 1 x ?
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Answers (1)

Daniel Valdez
Answered 2022-06-28 Author has 19 answers
Step 1
The inequality given to you looks like this
x - 2 x + 3 < x + 1 x
Right from the start, you know that any solution interval must not contain the values of x that will make the two denominators equal to zero.
More specifically, you need to have
x + 3 0 x - 3 and x 0
With that in mind, use the common denominator of the two fractions, which is equal to ( x + 3 ) x , to get rid of the denominators.
More specifically, multiply the first fraction by 1 = x x and the second fraction by
1 = x + 3 x + 3
This will get you
x - 2 x + 3 x x < x + 1 x x + 3 x + 3
x ( x - 2 ) x ( x + 3 ) < ( x + 1 ) ( x + 3 ) x ( x + 3 )
This is equivalent to
x ( x - 2 ) < ( x + 1 ) ( x + 3 )
Expand the parantheses to get
x 2 - 2 x < x 2 + x + 3 x + 3
Rearrange the inequality to isolate x on one side
- 6 x < 3 x > 3 ( - 6 ) x > - 1 2
This means that any value of x that is greater than - 1 2 , except x = 0 , will be a solution to the original inequality.
Therefore, the solution interval will be x ( - 1 2 , + ) \ { 0 }

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