Add or Subtract. `(3x)/(x^2+3x-10) - 4/(x-2)`

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lemjay | High School Teacher | (Level 3) Senior Educator

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`(3x)/(x^2+3x-10)-4/(x-2)`

Since the two fractions have unlike denominators, determine the LCD.

To do so, factor the denominator of the first fraction.

`=(3x)/((x+5)(x-2))-4/(x-2)`

Then, take the product of the different factors present in the denominators to get the LCD.
Since the different factors in the denominators are (x+5) and (x-2), then, the LCD is (x+5)(x-2).

So to express them with same denominators, multiply the second fraction by (x+5)/(x+5).

`=(3x)/((x+5)(x-2))-4/(x-2)*(x+5)/(x+5)`

`=(3x)/((x+5)(x-2))-(4(x+5))/((x+5)(x-2))`

Now that they have same denominators, proceed to subtract the two fractions.

`=(3x-4(x+5))/((x+5)(x-2))`

`=(3x-4x-20)/((x+5)(x-2))`

`=(-x-20)/((x+5)(x-2))`

`=(-(x+20))/((x+5)(x-2))`

`=-(x+20)/((x+5)(x-2))`

Hence,  `(3x)/(x^2+3x-10)-4/(x-2)=-(x+20)/((x+5)(x-2))` .

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