How do you simplify #((x)/(x+1))/(x+x/3)#?

Question

Kennedy Adams · Accepted Answer

#3/(4x+4)#

#(x/(x+1))/(x+x/3)#

First, add the terms in the denominator.

#=(x/(x+1))/((3x)/3+x/3)# (create a common denominator)

#=(x/(x+1))/((4x)/3)# (combine into a single fraction)

#=cancel(x)/(x+1)*3/(4cancel(x))# (rewrite expression and cancel)

#=3/(4(x+1))# (multiply fractions)

#=3/(4x+4)# (distribute in the denominator)

Nathan Axel · Answer

#3/(4x+4)#

Given:#" "(x/(x+1))/(x+x/3)#

Multiply by 1 but in the form of #3/3# giving:

#(x/(x+1))/(x+x/3)xx3/3" "=" "((3x)/(x+1))/(3x+x)" "=" "((3x)/(x+1))/(4x)#

This is the same as:

#(3x)/(x+1)xx1/(4x)" "=" "x/x xx3/4xx1/(x+1)" "=" "3/(4x+4)# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ #color(blue)("Check")#

Set #x=1#

#(x/(x+1))/(x+x/3)" "->" "1/2xx1/((4/3))" "=" "3/8#

#3/(4x+4)" "->" "3/8#

They match!

Hazel Anglin · Answer

#frac(3 x)(4 (x^(2) + x))#

#= 3/(4(x+1)# We have: #frac(frac(x)(x + 1))(x + frac(x)(3))# #= frac(x)(x + 1) times frac(1)(x + frac(x)(3))# #= frac(x)((x + 1)(x + frac(x)(3)))# #= frac(x)((x)(x) + (x)(frac(x)(3)) + (1)(x) + (1)(frac(x)(3))# #= frac(x)(x^(2) + frac(x^(2))(3) + x + frac(x)(3))# #= frac(x)(x^(2) (1 + frac(1)(3)) + x (1 + frac(1)(3)))# #= frac(x)(frac(4)(3) x^(2) + frac(4)(3) x)# #= frac(x)(frac(4)(3) (x^(2) + x))# #= frac(3 x)(4 (x^(2) + x))" "larr# factorise #=frac(3 x)(4x (x + 1))# #= 3/(4(x+1)#

Kennedy Adams · Answer

undefined To simplify ((x)/(x+1))/(x+x/3), we can follow these steps:

1. Simplify the expression within the parentheses: (x)/(x+1) becomes x/(x+1).
2. Simplify the expression x+x/3: To add x and x/3, we need a common denominator, which is 3. So, x+x/3 becomes (3x+x)/3, which simplifies to (4x)/3.

Now, we can rewrite the expression as x/(x+1) divided by (4x)/3.

To divide by a fraction, we multiply by its reciprocal. Therefore, we can rewrite the expression as x/(x+1) multiplied by 3/(4x).

Now, we can simplify further by canceling out common factors. The x in the numerator and denominator cancels out, leaving us with 1/(x+1) multiplied by 3/4.

So, the simplified expression is (3/4)/(x+1).