Question 287802
If {{{3/(x+1)+4/(x-2)=(px+q)/(x+1)(x-2)}}} must be true for every value of {{{x}}} ,
it will be true for {{{x=0}}} and for {{{x=1}}} .
 
For {{{x=0}}} :
{{{3/(0+1)+4/(0-2)=(px+q)/(0+1)(0-2)}}}
{{{3/1+4/(-2)=(p*0+q)/(1(-2))}}}
{{{3+(-2)=q/(-2)}}}
Multiplying times {{{(-2)}}} we get
{{{3(-2)+(-2)(-2)=q}}}-->{{{-6+4=q}}} --> {{{highlight(q=-2)}}}
 
For {{{x=1}}} :
{{{3/(1+1)+4/(1-2)=(p*1-2)/(1+1)(1-2)}}}
{{{3/2+4/(-1)=(p-2)/(2(-1))}}}
{{{3/2-4=(p-2)/(-2)}}}
{{{3/2-4=(p-2)/(-2)}}}
Multiplying times {{{(-2)}}} we get
{{{(3/2)(-2)-4(-2)=p-2}}}
{{{-3+8=p-2}}}
{{{8-3+2=p}}} --> {{{highlight(p=7)}}}