1. Simplify & state any excluded values

c-5
―---------
c²-25

2. Simplify & state any excluded values

x² - x - 6
----------------
x² + 7x + 10

9. Find the constant of variation, k, for y= –8 when
x = 12.

10. Write an inverse variation to model the situation
and answer the question. Two rectangular fields
have the same area. One measures 75 yd by 60 yd. If
the other has a length of 72 yd, what is its width?.

1. To simplify the expression (c-5)/(c²-25), we first factor the denominator as the difference of squares:

c²-25 = (c+5)(c-5)

Next, we can simplify the expression:

(c-5)/(c²-25) = (c-5)/[(c+5)(c-5)]

The (c-5) terms cancel out, leaving:

1/(c+5)

Therefore, the simplified expression is 1/(c+5). The excluded value is c = -5, as it would result in division by zero.

2. To simplify the expression (x² - x - 6)/(x² + 7x + 10), we can factor the numerator and denominator:

x² - x - 6 = (x-3)(x+2)
x² + 7x + 10 = (x+5)(x+2)

Now we can simplify the expression:

(x² - x - 6)/(x² + 7x + 10) = [(x-3)(x+2)]/[(x+5)(x+2)]

The (x+2) terms cancel out, leaving:

(x-3)/(x+5)

Therefore, the simplified expression is (x-3)/(x+5). The excluded value is x = -5, as it would result in division by zero.

9. The constant of variation, denoted as k, represents the ratio between two variables in a linear relationship. In this case, we are given that y = -8 when x = 12.

Using the formula for direct variation, y = kx, we can substitute the given values:

-8 = k(12)

Solving for k:

k = -8/12

Simplifying, we have:

k = -2/3

Therefore, the constant of variation, k, is -2/3.

10. Inverse variation occurs when the product of two variables is equal to a constant. In this case, the constant is the area of the rectangular fields.

The area of a rectangle is given by length multiplied by width.

For the first rectangular field with measurements 75 yd by 60 yd, the area is:

Area = 75 yd * 60 yd = 4500 yd²

Let's assume the width of the second rectangular field is represented by w. The length of the second field is given as 72 yd. So, the area of the second rectangular field can be expressed as:

Area = 72 yd * w

Since the areas of the two fields are equal, we can set up the inverse variation equation:

4500 yd² = 72 yd * w

To solve for the width, w, we divide both sides of the equation by 72 yd:

w = 4500 yd² / 72 yd

Simplifying the expression:

w = 62.5 yd

Therefore, the width of the second field is 62.5 yards.

1. To simplify the rational expression (fraction), we need to factor both the numerator and denominator.

First, let's factor the numerator, c - 5. This is already in its simplest form.

Next, let's factor the denominator, c² - 25. This is a difference of squares, which can be factored as (c + 5)(c - 5).

Now, we can rewrite the expression as (c - 5) / ((c + 5)(c - 5)).

The factor (c - 5) appears in both the numerator and denominator, so we can cancel them out.

The simplified expression is 1 / (c + 5).

However, there is an excluded value in this case. Since the denominator cannot be equal to zero, we have to find the value(s) of c that make the denominator equal to zero. In this case, when c = -5, the denominator becomes zero. Therefore, c = -5 would be an excluded value.

So the simplified expression is 1 / (c + 5), with an excluded value of c = -5.

2. To simplify this rational expression, we again need to factor both the numerator and denominator.

Let's factor the numerator, x² - x - 6. This factors as (x - 3)(x + 2).

Now, let's factor the denominator, x² + 7x + 10. This factors as (x + 2)(x + 5).

The simplified expression is (x - 3) / ((x + 2)(x + 5)).

The factor (x + 2) appears in both the numerator and denominator, so we can cancel them out.

The simplified expression is (x - 3) / (x + 5).

Now, let's check for any excluded values. In this case, there are no values of x that make the denominator equal to zero. Therefore, there are no excluded values.

So the simplified expression is (x - 3) / (x + 5), with no excluded values.

9. The constant of variation, denoted as k, represents the relationship between two variables in a direct or inverse variation equation.

In this case, we are given the equation y = -8 and x = 12.

To find the constant of variation, we can rearrange the equation to isolate k.

Start with the equation y = kx, where y is the dependent variable, k is the constant of variation, and x is the independent variable.

Substitute the given values into the equation: -8 = k * 12.

Now, solve for k. Divide both sides of the equation by 12: -8/12 = k.

Reduce the fraction -8/12 to -2/3: k = -2/3.

Therefore, the constant of variation, k, for the equation y = -8 when x = 12 is -2/3.

10. In an inverse variation, one variable increases as the other variable decreases, and vice versa.

To write an inverse variation equation, we can use the formula y = k/x, where y and x are the two variables, and k is the constant of variation.

In this situation, we know that the two rectangular fields have the same area. Let's denote the length of the second field as x and its width as y (which we need to find).

The area of a rectangle is given by length multiplied by width. Therefore, the first rectangle has an area of 75 * 60 = 4500 square yards.

Using the inverse variation formula, we can write the equation as 4500 = k/(72), where k is the constant of variation.

To find k, we can rearrange the equation: k = 4500 * 72.

Multiply 4500 by 72 to get the value of k.

Now that we have the value of k, we can use it to find the width (y) of the second rectangle when its length (x) is 72 yards.

Plug in the values into the inverse variation equation: y = k/(x).

Substitute the values of k and x: y = (4500 * 72) / (72).

Simplify the equation: y = 4500.

Therefore, the width of the second rectangular field is 4500 yards when its length is 72 yards, based on the given inverse variation.

For the first one.... if you factor the denominator you will be able to reduce the rational expression. Be certain to state the restrictions on the denominator before you reduce : )

Similarly for the second one... you must factor the top and bottom, then state restrictions then reduce : )
= (x-3)(x+2)/ (x+5)(x+2)
notice the x+2 that can reduce on top and bottom... but first state the restriction on the denominator... that is,
x+2 can not = 0
x can not = -2 and
x + 5 can not = 0
x can not = -5

your turn....