Expert: Abe Mantell - 7/24/2011

Question
27. Simplify: (x+3/ 5) - ( 2x+1/10)

28.  SImplify: (x+3/5) + (x-2/2x)

32. Simplify: 1/p^2 - 1/q^2
                    ______________
                     2/p^2 - 1/pq - 1/q^2

33. Simplify: x^-2 - x^2/ x^-1-x

34. Simplify: 1/x - 1/y
                    __________
                   y/x- x/y

Answer
Hello Anna,

Should the fractions be (x+3)/5 instead of x+3/5?  I think so...

27. Simplify: ((x+3)/5)-((2x+1)/10)
= (2(x+3)/10)-((2x+1)/10)
= ((2x+6)-(2x+1))/10
= (2x+6-2x-1)/10
= 5/10
= 1/2

28. Simplify: (x+3)/5 + (x-2)/(2x)
= 2x(x+3)/(10x) + 5(x-2)/(10x)
= (2x^2+6x + 5x-10)/(10x)
= (2x^2+11x-10)/(10x)

32. Multiply numerator and denominator by p^2*q^2, to get
q^2-p^2
---------- , which factors as
2q^2-pq-p^2

(q-p)(q+p)
---------- , which simplifies to
(2q+p)(q-p)

q+p
----
2q+p

33. Simplify: (x^-2 - x^2)/(x^-1-x), multiply by x^2/x^2
(1-x^4)/(x-x^3), which factors to
=(1-x^2)(1+x^2)/[x(1-x^2)]
=(1-x)(1+x)(1+x^2)/[x(1-x)(1+x)]
=(1+x^2)/x

34. Simplify: 1/x - 1/y
             _________
             y/x - x/y

multiply by (xy)/(xy) to get
(y-x)/(y^2-x^2)
=(y-x)/[(y-x)(y+x)]
= 1/(y+x)

OK?

Abe