How do you simplify #(x+5)/(x-5) div (x^2-25)/(5-x)#?

Question

Ethan Adkins · Accepted Answer

#-1/((x-5))# #(x+5)/(x-5) div(x^2-25)/(5-x)# #(x+5)/cancel((x-5))xx(-cancel((x-5)))/(x^2-25)# #-((x+5))/(x^2-25)# #"We can write as "-(x+5)/((x-5)(x+5))" so "(x^2-5^2)=(x-5)(x+5)# #-cancel(x+5)/((x-5)* cancel((x+5))# #-1/((x-5))#

Eli Assouline · Answer

#(-1)/(x-5) = 1/(5-x)#

The first step in most algebraic fraction problems is to find factors.

#(x+5)/(x-5) color(red)(div)(color(magenta)(x^2-25))/(5-x)" "color(magenta)(x^2-25)"( difference of squares)"#

=#(x+5)/(x-5) color(red)(xx)color(blue)((5-x))/(color(magenta)((x+5)(x-5))#

NOTE: # color(blue)((5-x) = -(x-5))#

=#cancel(x+5)/cancel(x-5) color(red)(xx)color(blue)((-cancel((x-5)^1))/(cancel((x+5))(x-5)#

=#(-1)/(x-5)#

=#1/(5-x)#

Kennedy Adams · Answer

undefined To simplify the expression (x+5)/(x-5) divided by (x^2-25)/(5-x), we can start by simplifying each fraction individually.

The first fraction (x+5)/(x-5) can be simplified by factoring the numerator and denominator.

The numerator (x+5) cannot be factored further, but the denominator (x-5) can be factored as (x-5) = -(5-x).

Now, let's simplify the second fraction (x^2-25)/(5-x).

The numerator (x^2-25) is a difference of squares and can be factored as (x+5)(x-5).

The denominator (5-x) can be rewritten as -(x-5).

Now, we can rewrite the expression as (x+5)/(-(5-x)) multiplied by (5-x)/(x+5).

The (x+5) terms cancel out, and we are left with -1/(x-5).

Therefore, the simplified expression is -1/(x-5).