Question

For how many integers x is the fraction
4x-17/5x +9
also an integer?

Answer 1

Answer:

values of x=-2, -4, -26

Step-by-step explanation:

Hint: if m = (4x-17)/(5x+9), then

5m = (20x-85)/(5x+9) = 4 - 121/(5x+9)

Answer 2

Answer:

3

Step-by-step explanation:

Let the given fraction is equals to integer k(say).

then,

$\frac{4x-17}{5x+9} =k \,\, where\,\,\, k \in \mathbb{Z}$

$\implies k = \frac{1}{5} \bigg[4-\frac{121}{5x+9} \bigg]$

Now since K is Integer , so 5 k wil also be integer. In RHS note that, there is a fraction term, if that term will be integer then its negative and addition with 4, doesn't harm its property of being integer, thus all criteria are met.

So, for that fractional term to be integer (5x+9) must divide 121 leaving remainder zero.

Also, note that only factors of 121 are $\pm 1, \pm 11, \pm 121.$

$5x+9 = 1 \,\, gives\,\, x\notin \mathbb{Z},\, \, but \,\,solving \,\, 5x+9 = 1 \,\, we \,\,get \\x= -2 (x \in \mathbb{Z}).$

Similarly,

$5x+9 = 11 \,\, gives\,\, x\notin \mathbb{Z},\, \, but \,\,solving \,\, 5x+9 =-1 1 \,\, we \,\,get \\x= -4 (x \in \mathbb{Z}).$

and

$5x+9 = 121 \,\, gives\,\, x\notin \mathbb{Z},\, \, but \,\,solving \,\, 5x+9 = -121 \,\, we \,\,get \\x= -26 (x \in \mathbb{Z}).$

So, finally we are getting 3 possible integer values for x

as x= -2,-4, and-26.