Please help with these 5 problems!

I have the solutions but I don't understand how to get them!

1. Perform the indicated operations and simplify:
[(2/x)-1]/(x^2 -4)

2. Subtract and simplify:
[12/(x^2 -4)] - [(3-x)/(x^2 + 2x)]

3. Perform the indicated operations and simplify:
[(x/y)-(y/x)]^-1

4. Divide and simplify:
[(a^2 b)/(a-b)]/[(a+b)/(a^2-b^2)]

5. Perform the indicated operations and simplify
[(x/x-3)-(2x/x^2-2X-3)]/[(2/x+1)-(1/x)

Solutions:

1. -1/(x^2 + 2x)

2. [x^2 + 7x + 6]/[x(x+z)(x-z)]
--typo in book?

3. xy/(x^2 - y^2)

4. a^2 b

5. x^2/(x-3)

1. Rewrite (2/x) -1 in the numerator as -(x-2)/x

Factor (x^2/4) in the denominator as (x-2)(x+2)
Then cancel out the (x-2) terms in numerator and denominator. You wil be left with -1/[x(x+2)]

3. Rewrite x/y and y/x with a common denominator, xy.
[(x^2- y^2)/xy]^-1 = xy/(x^2- y^2)

You try the others. They are all exercises in factoring and canceling tems

8/9a^2 divide 4a^2-4a-24/a^2-6a+9 Perform the indicated operation

(2/3-1/6)divide by(1/4+4/5) reduce to lowes terms

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Sure, I can help you understand how to solve these problems step by step. Let's go through each problem one by one.

1. To perform the indicated operations and simplify [(2/x)-1]/(x^2 -4):
First, simplify the numerator by finding a common denominator for 2 and x. The common denominator is x, so rewrite 2 as 2x/x. Then, subtract 1 from 2x/x to get: (2x/x -1)/ (x^2 - 4). Next, simplify the numerator further by combining the terms: (2x - x)/x / (x^2 - 4). This simplifies to x/x / (x^2 - 4). Finally, simplify x/x to 1 and the expression becomes 1 / (x^2 - 4).

2. To subtract and simplify [12/(x^2 -4)] - [(3-x)/(x^2 + 2x)]:
First, find a common denominator for both fractions, which is (x^2 - 4)(x^2 + 2x). Rewrite the fractions with the common denominator: (12(x^2 + 2x) - (3-x)(x^2 - 4)) / ((x^2 - 4)(x^2 + 2x)). Next, simplify the numerator by performing the multiplication and subtraction: (12x^2 + 24x - (3x^3 - 13x^2 + 4x + 12)) / ((x^2 - 4)(x^2 + 2x)). This simplifies to (-3x^3 + 25x^2 + 20x - 12) / ((x^2 - 4)(x^2 + 2x)).

3. To perform the indicated operations and simplify [(x/y)-(y/x)]^-1:
First, simplify the expression inside the square brackets: (x/y) - (y/x). To find a common denominator, multiply the first fraction by x/x and the second fraction by y/y: (x^2/xy) - (y^2/xy). Combine the terms: (x^2 - y^2) / xy. Finally, take the reciprocal of this expression by flipping the numerator and denominator: xy / (x^2 - y^2).

4. To divide and simplify [(a^2 b)/(a-b)] / [(a+b)/(a^2-b^2)]:
Dividing fractions is equivalent to multiplying by the reciprocal of the second fraction. So, rewrite the expression as [(a^2 b)/(a-b)] * [(a^2-b^2)/(a+b)]. Next, simplify both the numerator and denominator. In the numerator, (a^2 b) cancels out with (a^2-b^2), leaving just b. In the denominator, (a-b) cancels out with (a+b), leaving just 1. Therefore, the simplified expression is b.

5. To perform the indicated operations and simplify [(x/x-3)-(2x/x^2-2x-3)] / [(2/x+1)-(1/x)]:
First, simplify the expression inside the first square brackets: (x/(x-3)) - (2x/(x^2 - 2x - 3)). To find a common denominator, multiply the first fraction by (x+1)/(x+1) and the second fraction by x/x: (x(x+1)/(x(x-3))) - (2x/(x(x-3))). Combine the terms in the numerator: (x^2 + x)/(x(x-3)) - (2x/(x(x-3))). To simplify further, find the common denominator of x(x-3), which is x(x-3). Multiply the first fraction by (x)/(x) and the second fraction by (x-3)/(x-3): (x(x^2 + x)/(x(x-3)(x))) - (2x(x-3)/(x(x-3)(x-3))). Combine the terms in the numerator: (x^3 + x^2 - 2x^2 + 6x)/(x(x-3)(x-3)). Finally, simplify the numerator: (x^3 - x^2 + 6x)/(x(x-3)(x-3)).

These step-by-step explanations should help you understand how to solve the given problems.