How do you solve x/(x-2)-1/(x-3)=(1)?

2 Answers
Oct 7, 2017

x = 4

Explanation:

Multiplying the numerators of the fractions by
the common denominator, (x-2)(x-3),

(x(x-3)-1(x-2))/((x-2)(x-3)) = 1

Multiplying both sides by (x-2)(x-3),

implies (x^2 - 3x - x + 2) = x^2-5x+6

implies -4x + 5x = 6 - 2

implies x = 4

x=4

Explanation:

The first thing I'd do is combine the two fractions on the left:

x/(x-2)(1)-1/(x-3)(1)=1

x/(x-2)((x-3)/(x-3))-1/(x-3)((x-2)/(x-2))=1

(x^2-3x)/((x-2)(x-3))-(x-2)/((x-2)(x-3))=1

(x^2-4x+2)/(x^2-5x+6)=1

And now multiply through by the denominator of the fraction:

x^2-4x+2=x^2-5x+6

We can now simplify:

x^2color(red)(-x^2)-4xcolor(red)(+5x)+2color(red)(-2)=x^2color(red)(-x^2)-5xcolor(red)(+5x)+6color(red)(-2)

x=4