How do you simplify the expression #(b/(b+3)+2)/(b^2-2b-8)#?

Answer 1

#3/(b^2-b-12)#

Given:#" "(b/(b+3)+2)/(b^2-2b-8 )#
#color(blue)("Consider the numerator: "b/(b+3)+2/1)#
I have deliberately chosen to write 2 as #2/1#. Although correct this is not normally done.
#color(green)([b/(b+3)]+[2/1color(red)(xx1)])#
#color(green)([b/(b+3)]+[2/1color(red)(xx(b+3)/(b+3))])#
#[b/(b+3)]+[(2b+6)/(b+3)]#
#(3b+6)/(b+3)# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ #color(blue)("Consider the denominator: "b^2-2b-8)#
Notice that #2xx(-4)=-8 and 2-4=-2#
Write as #(b-4)(b+2)# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ #color(blue)("Putting it all together")#
numerator#-:# denominator#->(3b+6)/(b+3)-:(b-4)(b+2)#
#" "=" "(3b+6)/(b+3)xx1/((b-4)(b+2))#
Factor out the 3 from #3b+6->3(b+2)#
#" "=" "(3cancel((b+2)))/(b+3)xx1/((b-4)cancel((b+2)))#
#" "=" "3/(b^2-b-12)#
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Answer 2

To simplify the expression (b/(b+3)+2)/(b^2-2b-8), you can follow these steps:

  1. Factor the denominator (b^2-2b-8) as (b-4)(b+2).
  2. Rewrite the expression as b/(b+3) + 2/(b-4)(b+2).
  3. Find a common denominator for the fractions, which is (b+3)(b-4)(b+2).
  4. Multiply the first fraction by (b-4)/(b-4) and the second fraction by (b+3)/(b+3) to get (b(b-4))/((b+3)(b-4)(b+2)) + (2(b+3))/((b+3)(b-4)(b+2)).
  5. Combine the fractions by adding the numerators: (b(b-4) + 2(b+3))/((b+3)(b-4)(b+2)).
  6. Simplify the numerator: (b^2 - 4b + 2b + 6)/((b+3)(b-4)(b+2)).
  7. Combine like terms in the numerator: (b^2 - 2b + 6)/((b+3)(b-4)(b+2)).
  8. The simplified expression is (b^2 - 2b + 6)/((b+3)(b-4)(b+2)).
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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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