Simplify the expressions and solve the equations

Question

1. 1 over t-1 minus 3 over t-3 both equal 1/4

 

2. 4x² - x - 14 over x-2 divided by 4x² + x-14 over x+2

Answer 1

4x² - x - 14  ÷  4x² + x - 14    

     x - 2                  x + 2

 

dividing by a fraction is the same a multiplying by the reciprocal of the fraction.

So we will change this to multiplication, and factor the quadratics

 

(4x + 7)(x - 2) (4x - 7)(x + 2)

        (x - 2)              (x + 2)

 

(4x + 7)(4x - 7) 

 

16x² - 49

Answer 2

Nkeiruka:

 

 

1. I take your "both equal" as follows

1 / (t - 1) - 3 / (t - 3) = 1 / 4.

 

The LCM on the left is (t-1)(t-3).  The equation is

 

[t - 3 - 3(t - 1)] / (t-1)(t-3)  = 1 / 4.  Never divide by zero, so assume that t is not equal to 1 or t is not equal to 3, or both.

 

Now simplify the above expression as

 

 

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

 

Cross-multiply and multiply out the factors.

 

- 8t = t² - 4t + 3

 

You get

 

t² + 4t + 3 = 0.

 

Factorize.

 

(t +1) ( t + 3) = 0.

 

So the roots are either t = -1 or t = -3.

 

 

Your Number 2 is NOT an equation, it is just an expression, not equated to anything.  Incomplete?

 

 

Dattaprabhakar (Dr. G.)