Question

(x-1/x ²-4) + (1/x-2)

Answer 1

Answer:

  (2x+1)/(x^2 -4)

Step-by-step explanation:

When you add fractions (or rational expressions) you need to express each of them over a common denominator, so you can add the numerators. (Parentheses or other grouping symbols are required.)

  $\dfrac{x-1}{x^2-4}+\dfrac{1}{x-2}=\dfrac{x-1}{(x-2)(x+2)}+\dfrac{(1)(x+2)}{(x-2)(x+2)} \qquad\text{now we have a common denominator}\\\\=\dfrac{(x-1)+(x+2)}{(x-2)(x+2)}=\dfrac{2x+1}{(x-2)(x+2)}=\dfrac{2x+1}{x^2-4}$

You factor the denominator on the left so you can see what factors are missing from the denominator on the right.

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Numerical example

Suppose you want to add the fractions 4/15 and 3/5. You would factor the denominator 15 into 3·5, so you can see that the fraction 3/5 needs to be multiplied by (3/3) in order to give it a denominator of 15. Once the two fractions have a common denominator, you can add their numerators.

  4/15 + 3/5 = 4/15 + (3/5)(3/3)

  = 4/15 + 9/15

  = (4+9)/15 = 13/15

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Comment on grouping symbols

When a rational expression is typeset, the division bar of the fraction identifies the numerator and the denominator:

  $\dfrac{x-1}{x^2-4}$

When you write it in plain text as ...

  x - 1/x^2 -4

this becomes a 3-term expression:

  (x) -(1/x^2) -(4)

which is very different from what you may intend. In order to match the original typeset expression, parentheses must be added to show the group in the numerator and the group in the denominator:

  (x -1)/(x^2 -4)

The same holds for exponents. The superscript position in a typeset expression tells you what the exponent is. When you write the same thing in plain text, parentheses are needed to show the exponent group.

  $3^{x-1}$ = 3^(x -1) ≠ 3^x -1