sums and differences of a rational algebraic expression 3/a + 2/ab + 1/abc another is 2x-1/6 + x+3/12 another is 2/a-b + 3/a+b another is 2a+b/a+b - 2a-b/a-b - 3a^2-b^2/a^2-b^2 another is 2x/2x^2+5x-3 + 5/x^2-9 another is x-3/12x^3-18x^2 - x+1/16x^2-24x

To find the sums and differences of rational algebraic expressions, follow the steps outlined below:

1. Start by finding a common denominator for the fractions involved in the expression. This step is crucial for simplifying the expression and performing addition or subtraction.

2. Add or subtract the numerators while keeping the common denominator unchanged.

Let's apply these steps to the expressions you provided:

Expression 1: 3/a + 2/ab + 1/abc

Step 1: The common denominator for these fractions is abc.

Step 2: Rewrite each fraction with the common denominator:
(3 * bc)/(abc) + (2 * ac)/(abc) + (1 * ab)/(abc)

Step 3: Combine the numerators:
(3bc + 2ac + ab)/(abc)

Expression 2: (2x-1)/6 + (x+3)/12

Step 1: The common denominator for these fractions is 12.

Step 2: Rewrite each fraction with the common denominator:
((2x - 1) * 2)/(12) + ((x + 3) * 1)/(12)

Step 3: Combine the numerators:
(4x - 2 + x + 3)/(12)

Simplify: (5x + 1)/(12)

Expression 3: 2/(a-b) + 3/(a+b)

Step 1: The common denominator for these fractions is (a+b)(a-b).

Step 2: Rewrite each fraction with the common denominator:
(2 * (a+b))/((a-b)(a+b)) + (3 * (a-b))/((a+b)(a-b))

Step 3: Combine the numerators:
(2a + 2b + 3a - 3b)/((a+b)(a-b))

Simplify: (5a - b)/((a+b)(a-b))

Expression 4: (2a + b)/(a+b) - (2a - b)/(a-b) - (3a^2 - b^2)/(a^2 - b^2)

Step 1: The common denominator for these fractions is (a+b)(a-b).

Step 2: Rewrite each fraction with the common denominator:
((2a + b)(a-b))/((a+b)(a-b)) - ((2a - b)(a+b))/((a-b)(a+b)) - ((3a^2 - b^2))/((a^2 - b^2))

Step 3: Expand and simplify each numerator:
(2a^2 - 2ab + ab - b^2)/((a^2 - b^2)) - (2a^2 - 2ab + ab + b^2)/((a^2 - b^2)) - (3a^2 - b^2)/((a^2 - b^2))

Step 4: Combine like terms and simplify:
(-3a^2 - 2b^2)/((a^2 - b^2))

Expression 5: (2x)/(2x^2 + 5x - 3) + (5)/(x^2 - 9)

Step 1: The denominators do not have a common factor, so we cannot simplify the expression further.

No further steps can be taken until we know the exact form or operation required for the expression x-3 over 12x^3 - 18x^2 - x + 1 over 16x^2 - 24x. Please provide more information if you have any specific requirements for this expression.

Remember, always simplify further if possible and check for any restrictions on the variables that may cause division by zero in the denominator.