Question

a number is divided by a quantity four less than itself, represented by the expression below. which values of x will result in a quantity less than one? greater than one? which values of x cannot be used at all, and why?

Answer 1

For the quantity less than one, all the x-values smaller than 4

For the quantity greater than one, all the x-values larger than 4

the value 4 cannot be used at all.

Explanation:

According to the description, the quantity in question can be represented by the fraction:

`(x)/(x-4)`

Notice that since the binomial (x - 4) is in the denominator, in order to prevent a case with undefined quotient, x - 4 cannot be zero, and that is x cannot be 4.

Notice as well that in the case that x is larger than 4, the binomial (x-4) is a positive number, and in the case that x is less than 4, the binomial (x - 4) is a negative number.

Which values result in the quantity greater than one?

We need to solve for x in the inequality:

`(x)/(x-4) >1`

So, if x > 4 then we can proceed as follows:

`(x)/(x-4) >1\\x>x-4\\0>-4`

which is a true statement, when x > 4

If x < 4 then:

`(x)/(x-4) >1\\x<x-4\\0<-4`

where we have used that (x-4) is negative, so multiplying by it would flip the direction of the inequality. As we see, this case results in an absurd , so it is not possible for x < 4 to render the quantity under study larger than one.

We study similarly the case for the quantity in question being smaller than one considering x > 4:

`(x)/(x-4) <1\\x<x-4\\0<-4`

and we arrive at an absurd. so the quantity cannot be smaller than 1 if x is larger than 4

Now for x smaller than 4:

`(x)/(x-4) <1\\x>x-4\\0>-4`

we arrive at a true statement. So it is possible to get the quantity in question smaller that one if x is less than 4.