Could someone help with the following math question? Find x: (2x +5) / (x+4)<1 

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Wilson2014 | eNotes Employee

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To solve for `(2x+5)/(x+4) < 1` , it is important to avoid multiplying/dividing a factor with "x" in it because "x" is unknown, and if it is a negative value, multiplying/dividing by it could cause the inequality sign to switch from < to >. Therefore, to solve this, subtract 1 from both sides.

`(2x+5)/(x+4) - 1 < 0 = (2x+5)/(x+4) - (x+4)/(x+4) < 0`

(We converted 1 to a fraction form so that we can combine like terms.)

`rArr (2x-x+5-4)/(x+4)<0`

`rArr (x+1)/(x+4)<0`

Next, find the critical points by changing the inequality sign to an equals sign and try to solve for x. Also, any x value that makes the equation undefined is also a critical point.

`(x+1)/(x+4) = 0`

This statement is true when x=-1. This statement is undefined when x=-4 because the denominator would equal zero. Next, we take these critical points and test points that are between them on the number line and also points around these critical values.

For example, if we take a number between -4 and -1, such as -3, the inequality would solve to be


This is true. So we can conclude that all values of x between -4 and -1 would make the inequality true.

Next, try a value less than -4 and greater than -1. For example, -5 and 0.

`x=-5rArr 4<0` which is false, so anything less than -4 is not a solution for x. 

`x=0rArr 1/4<0` which is also false, so anything greater than -1 is not a solution for x. 

Using the same method, the critical values x=-1 and x=-4 are also not part of the solution.


Therefore, the solution to the inequality {x| -4<x<-1}.                        

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