1. Subtract and simplify

18/y-8 - 18/8-y
2. subtract simplify if possible
v-5/v - 7v-31/11v

3. Simplify by removing factors of 1
r^2-81/(r+9)^2

4. Divide and simplify
v^2-16/25v+100 divided by v-4/20

5. find all rational numbers for which the rational expression is undefined:
-18/7w

6. solve:
2/Z+4= 1/z-6

1. 18 / (y - 8) - 18 / (8 - y),

Multiply numerator and denominator of
2nd fraction by -1 and get:

18 / (y - 8) + 18 / (y - 8) =
36 / (y - 8).

2. (v - 5) / v - (7v - 31) / 11v,
Common denominator = 11v:
(11v - 55 -7v + 31) / 11v,
Combine like-terms:
(4v - 24) / 11v = 4(v - 6) / 11v.

3. (r^2 - 81) / (r + 9)^2,
Factor the numerator:
(r + 9) (r - 9) / (r + 9)^2,
Multiply numerator and denomjnator by
(r + 9)^2 and get:

(r + 9)^-1 (r - 9) / 1,
Multiply numerator and denominator by
(r + 9)^1:

(r - 9) / (r + 9).

4. (v^2 - (16/25)v + 100) / (v - 4)/20
Multiply numerator and denominator by 25 and get:
(V^2 - 16V + 100) / 5(V - 4) / 4,

Multiply numerator and denominqator by
4/5 and get:
4/5(v^2 - 16v + 100) / (v - 4).

6. 2 / (z + 4) = 1 / (z - 6),
Cross multiply:
2z - 12 = z + 4,
2z - z = 4 + 12,
z = 16.

_4y_____ _ ___3y________

y^2+6y+5 y^2+5y+4

1. To subtract and simplify the expression 18/y - 8 - 18/(8 - y), we need to find a common denominator. The common denominator in this case is y(8 - y).

First, we multiply the numerator and denominator of the first fraction by (8 - y):
(18/y) * (8 - y)/(8 - y) = (18(8 - y))/(y(8 - y)) = (144 - 18y)/(y(8 - y))

Next, we multiply the numerator and denominator of the second fraction by y:
(18/(8 - y)) * y/y = (18y)/(y(8 - y))

Now that the denominators are the same, we can combine the numerators:
(144 - 18y)/(y(8 - y)) - (18y)/(y(8 - y)) = (144 - 18y - 18y)/(y(8 - y)) = (144 - 36y)/(y(8 - y))

This is the simplified result.

2. To subtract and simplify the expression (v - 5)/v - (7v - 31)/(11v), we need to find a common denominator. The common denominator in this case is 11v.

First, we multiply the numerator and denominator of the first fraction by 11:
[(v - 5)/v] * 11/11 = [(11v - 55)/(11v)]

Next, we multiply the numerator and denominator of the second fraction by v:
[(7v - 31)/(11v)] * v/v = [(7v^2 - 31v)/(11v^2)]

Now that the denominators are the same, we can combine the numerators:
[(11v - 55)/(11v)] - [(7v^2 - 31v)/(11v^2)] = [(11v^2 - 55v - 7v^2 + 31v)/(11v^2)] = [(4v^2 - 24v)/(11v^2)]

This is the simplified result.

3. To simplify the expression (r^2 - 81)/(r + 9)^2 by removing factors of 1, we factor the numerator and denominator.

The numerator is a difference of squares and can be factored as (r + 9)(r - 9).

The denominator is already factored as (r + 9)(r + 9) or (r + 9)^2.

Now, we can remove the common factor of (r + 9):
(r + 9)(r - 9)/(r + 9)(r + 9) = (r - 9)/(r + 9)

This is the simplified result.

4. To divide and simplify the expression (v^2 - 16)/(25v + 100) divided by (v - 4)/20, we can convert it to multiplication by taking the reciprocal of the second fraction.

The reciprocal of (v - 4)/20 is 20/(v - 4).

Now, we can rewrite the expression as:

[(v^2 - 16)/(25v + 100)] * [20/(v - 4)]

Next, factor the numerator as a difference of squares:

[(v + 4)(v - 4)/(25v + 100)] * [20/(v - 4)]

Now, we can cancel out the common factor of (v - 4):

[(v + 4)/25] * [20/1] = (v + 4)/5

This is the simplified result.

5. To find all rational numbers for which the rational expression -18/7w is undefined, we need to identify the values of w that would result in a denominator of zero.

In this case, the expression is undefined when the denominator 7w equals zero. So we need to solve the equation 7w = 0:

7w = 0
w = 0/7
w = 0

Therefore, the rational expression is undefined when w = 0.

6. To solve the equation 2/(z + 4) = 1/(z - 6), we can first cross-multiply to eliminate the denominators.

Cross-multiplying gives us:

2(z - 6) = 1(z + 4)

Next, distribute and simplify:

2z - 12 = z + 4

Now, we can isolate the variable z by subtracting z from both sides and adding 12 to both sides:

2z - z = 4 + 12

Simplifying:

z = 16

Therefore, the solution to the equation is z = 16.