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Question 1.

There are numbers a and b for which

$$\frac A{x-1}+\frac B{x+1}=\frac{x+2}{x^2-1}$$

for every number $$x\neq\pm1$$. Find b.

Question 2.

If h>0, find all values of h such that $$\lceil h\rceil\cdot\lfloor h\rfloor=72$$. Express your answer using interval notation.

Nov 28, 2020

#1
+1

(1)    Note  x^2 - 1  = (x - 1) (x + 1)

x + 2                        A                 B

____________   =    _______  +   _____                      multiply through by (x - 1) ( x + 1)

(x - 1) ( x + 1)               x - 1            x + 1

x + 2 =   A ( x + 1)  +  B ( x - 1)      simplify  as

1x + 2 =  (A + B)x  + (A - B)

Equating terms we have that

A + B  =  1

A - B  = 2      add these equations

2A  = 3

A = 3/2

And

3/2 + B = 1

B = -1/2   Nov 28, 2020
edited by CPhill  Nov 28, 2020
#2
+1

Thanks so much for all the help. I read your solution and I think you did everything great exept you kinda got a and b mixed up.

Nov 28, 2020
#4
0

Oops....let me fix that  !!!   CPhill  Nov 28, 2020
edited by CPhill  Nov 28, 2020
edited by CPhill  Nov 28, 2020
#3
+1

2)  Note  that

9  * 8  =  72

So  ceiling  8....  = 9

And floor  8.....  =   8

This indicates  that  the solution is

(  8 < h < 9 )   Nov 28, 2020