SAT QUESTIONS ON RATIONAL EXPRESSIONS

Problem 1 :

If y > 5, which of the following is equivalent to 

1 / [(1/(y - 4) + 1/(y - 3)]

a)  2y - 7    b) y² - 7y + 12       c)  y² - 7y + 12/(2y - 7)      d)  (2y - 7) / y² - 7y + 12

Solution :

= 1 / [(1/(y - 4) + 1/(y - 3)] -------(1)

Simplifying the denominator :

= [(1/(y - 4) + 1/(y - 3)]

= [(y - 3) + (y - 4)] / (y - 3)(y - 4)

= (2y - 7)/y² - 7y + 12

Applying this denominator in (1), we get

= 1 / [(2y - 7)/(y² - 7y + 12)]

= (y² - 7y + 12) / (2y - 7)

Option c is correct.

Problem 2 :

For what vaue of x is the expression

2 / [(x - 6)² + 4(x - 7) + 8]

undefined ?

Solution :

= 2 / [(x - 6)² + 4(x - 7) + 8]

When the denominator becomes 0, the function will become undefined.

(x - 6)² + 4(x - 7) + 8 = 0

Let x - 7 = t

(x - 6) - 1 = t

x - 6 = t + 1

Applying the values of x - 6 and x - 7, we get 

(t + 1)² + 4t + 8 = 0

t² + 2t + 1 + 4t + 8 = 0

t² + 6t + 9 = 0

(t + 3)(t + 3) = 0

t = -3 and t = -3

So, when t = -3 the given function will become undefined.

Problem 3 :

If 7/y = 17/(y + 30), what is the value of y/7 ?

a)  1/3       b)  3       c) 7        d)  21

Solution :

7/y = 17/(y + 30)

7(y + 30) = 17y

7y + 210 = 17y

7y - 17y = -210

-10y = -210

y = 210/10

y = 21

y/7 = 21/7

= 3

So, the value of y/7 is 3, option b is correct.

Problem 4 :

(80x² + 84x - 13)/(kx - 4) = -16x - 4 - [29/(kx - 4)]

Teh equation above is true for all values of x ≠ 4/k, where k is constant. What is the value of k ?

a)  -5       b)  -2       c) 2        d)  5

Solution :

(80x² + 84x - 13)/(kx - 4) = -16x - 4 - [29/(kx - 4)] ----(1)

In the left side, dividing the numerator by denominator is not possible. Because we have unknwon at the denominator.

Simplifying right side is possible.

= -16x - 4 - [29/(kx - 4)]

= [(-16x - 4)(kx - 4) - 29]/(kx - 4)

= [(-16kx² + 64x - 4kx + 16) - 29] / (kx - 4)

= [-16kx² + (64 - 4k)x - 13] / (kx - 4)

Applying the above value in (1), we get

(80x² + 84x - 13)/(kx - 4) = [-16kx² + (64 - 4k)x - 13] / (kx - 4)

Equating the numerators, we get 

80x² + 84x - 13 = -16kx² + (64 - 4k)x - 13

Equating the corresponding terms, we get

80 = -16k and 84 = 64 - 4k

4k = 84 - 64

4k = 20

k = 20/4

k = 5

The value of k is 5, option d is correct.

Problem 5 :

If (x + y)/x is equal to 6/5, which of the following is true ?

a)  y/x = 1/5          b)  y/x = 11/5       c)  (x + y) / x = 1/5       d)  (x - 2y)/x = -1/5

Solution :

(x + y)/x = 6/5

Doing the possible simplification, we get

5(x + y) = 6x

5x + 5y = 6x

5x - 6x = -5y

-1x = -5y

x/y = 5/1

Then, y/x = 1/5

So, option a is correct.

Problem 6 :

Which of the following must be true if (t + u)/t = 12/11 ?

a)  u/t = 1/11          b)  u/t = 23/11      c)  (t - u) / t = 1/11       d)  (t + 2u)/t = -8/11

Solution :

(t + u)/t = 12/11

11(t + u) = 12t

11t + 11u = 12t

Combining the like terms, we get

11u = 12t - 11t

11u = 1t

t/u = 11/1

 u/t = 1/11

Option a is correct.

Problem 7 :

The equation

(36y² + 43y - 25)/(ky - 3) = (-9y - 4) - [37/(ky - 3)]

is true for all values of y ≠ 3/k, where k is constant, what is the value of k ?

a)  27          b)  4     c)  -4       d)  -27

Solution :

(36y² + 43y - 25)/(ky - 3) = (-9y - 4) - [37/(ky - 3)] ------(1)

Simplifying the right side :

= (-9y - 4) - [37/(ky - 3)]

= [(-9y - 4)(ky - 3) - 37]/(ky - 3)]

Using distributive property, we get

= (-9ky² + 27y - 4ky + 12 - 37)/(ky - 3)

= (-9ky² + (27 - 4k)y -25)/(ky - 3)

Applying the above in (1), we get

(36y² + 43y - 25)/(ky - 3) = (-9ky² + (27 - 4k)y -25)/(ky - 3)

36y² + 43y - 25 = -9ky² + (27 - 4k)y -25

By equating the corresponding terms, we get

-9k = 36

k = -4

So, option c is correct.

Problem 8 :

(4c + 1)/(2c - 3)² - 2/(2c - 3)

The expression above is equivalent to x/(2x - 3)², where x is a positive constant and c ≠ 3/2, what is the value of x ?

Solution :

= (4c + 1)/(2c - 3)² - 2/(2c - 3)

LCM of the denomiantor is (2c - 3)²

= [(4c + 1) - 2(2c - 3)] / (2c - 3)²

= [4c + 1 - 4c + 6] / (2c - 3)²

= 7 / (2c - 3)²

7 / (2c - 3)² =  x/(2x - 3)²

x = 7