How do you solve #\frac { 7x  49} { 3x ^ { 2} + 6x  72} + \frac { 1} { x + 6} = \frac { 5} { 3x  12}#?
First factor the terms in the denominators:
now cancel out and multiply through using the distributive property:
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To solve the equation ( \frac{7x  49}{3x^2 + 6x  72} + \frac{1}{x + 6} = \frac{5}{3x  12} ):
 Factor the denominators of the fractions, if possible.
 Find a common denominator for the fractions.
 Combine the fractions into one.
 Solve for ( x ) by simplifying and rearranging the equation.
 Check for extraneous solutions.
Let's go through these steps:

Factor the denominators of the fractions: [ 3x^2 + 6x  72 = 3(x^2 + 2x  24) = 3(x + 6)(x  4) ]

Find a common denominator: It is ( (3x + 6)(x  4) ).

Rewrite the equation with the common denominator: [ \frac{7x  49}{(3x + 6)(x  4)} + \frac{1}{x + 6} = \frac{5}{3(x  4)} ]

Combine the fractions: [ \frac{(7x  49) + (1)(3x + 6)}{(3x + 6)(x  4)} = \frac{5}{3(x  4)} ] [ \frac{7x  49 + 3x + 6}{(3x + 6)(x  4)} = \frac{5}{3(x  4)} ] [ \frac{10x  43}{(3x + 6)(x  4)} = \frac{5}{3(x  4)} ]

Cross multiply and solve the resulting equation: [ (10x  43)(3(x  4)) = 5(3x + 6)(x  4) ] [ 30x^2  160x + 171 = 15x^2  30x  60x + 120 ] [ 15x^2  130x + 51 = 0 ]
Now, you can solve this quadratic equation using factoring, completing the square, or the quadratic formula to find the values of ( x ).
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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