Choice a is equivalent because we can say that technically we are multiplying two fractions together: (xy)/z and (5(x + y))/1.  We multiply the numerators together and the denominators together and end up with xy (5x + 5y)/z.  xy (5y + 5x)/z is also equivalent because it is only simplifying what is inside the parentheses and switching the order- the commutative property tells us this is still the same expression.  5x²y + 5xy²/z is equivalent as it is just a simplified version when the numerators are multiplied out.  Choice 5x² + y²/z is not equivalent because it does not account for all the variables that were in the given expression and it does not use FOIL correctly.

First, we need to find all of the values that are in the set S, and then we need to find the values in T. Once we do this, we must find the numbers in the intersection of S and T, which means we must find the values contained in BOTH sets S and T.

S contains all of the values of x such that 2x + 4 < 8. We need to solve this inequality.

2x + 4 < 8

Subtract 4 from both sides.

2x < 4

Divide by 2.

x < 2

Thus, S contains all of the values of x that are less than (but not equal to) 2. 

Now, we need to do the same thing to find the values contained in T.

-2x + 3 < 8

Subtract 3 from both sides.

-2x < 5

Divide both sides by -2. Remember, when multiplying or dividing an inequality by a negative number, we must switch the sign.

x > -5/2

Therefore, T contains all of the values of x that are greater than -5/2, or -2.5.

Next, we must find the values that are contained in both S and T. In order to be in both sets, these numbers must be less than 2, but also greater than -2.5. Thus, the intersection of S and T consists of all numbers between -2.5 and 2.

The question asks us to find the sum of the integers in the intersection of S and T. This means we must find all of the integers between -2.5 and 2.

The integers between -2.5 and 2 are the following: -2, -1, 0, and 1. We cannot include 2, because the values in S are LESS than but not equal to 2. 

Lastly, we add up the values -2, -1, 0, and 1. The sum of these is -2.

The answer is -2. 

Let  be the number of pizzas made and sold. Each pizza will require $2 worth of ingredients, so the ingredients in total will cost . Add this to the cost to rent the equipment and the cost will be .

The pizzas will cost $5.50 each, so the money raised will be .

The profit will be the difference between the revenue and the cost - 

The Spanish club wants a profit of at least $600, so we set up and solve the inequality:

The Spanish club must sell at least 258 pizzas to earn a profit.

* Notice that when we multiply or divide both sides by a negative number the ineqaulity sign changes orientation.

Begin by moving all of the  values to the left side of the inequality:

becomes

Next, move the  to the right side:

Finally, divide both sides by :

First, move the  values to the left side of the inequality:

becomes

Next, move the  to the right side:

Finally, divide by . Remember: you must flip the inequality sign when you multiply or divide by a negative number.

First, get the  factors on the left side of the inequality:

becomes

Next, subtract  from both sides:

Now, divide by .  Remember: Dividing or multiplying by a negative number requires you to flip the inequality sign:

First, multiplying each side of the equality by  gives . Next, dividing each side of the inequality by  will solve for ; .

We simplify this inequality similarly to how we would simplify an equation

Thus 

In order to find the solution set, we solve  as we would an equation:

Therefore, the solution set is any value of .