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. 

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 .

Recall that when you have an absolute value and an inequality like

,

this is the same as saying that  must be between  and .  You can rewrite it:

To solve this, you just apply your modifications to each and every part of the inequality.

First, subtract :

Then, divide by :

Next, do the same for the other equation.

becomes...

Then, subtract :

Then, divide by :

Thus, the smallest value for  is , while the smallest value for  is .  Therefore, quantity A is greater.

If each pencil sells at 50 cents,  pencils will sell at . The smallest value of  such that

The answer is: 

From the equation we can see that the slope is –1.

Take the values of y that are possible, i.e. 2 and 3, and plug them into the first inequality. First, plug in 2. 2 – 3x > 21. Subtract 2 from both sides, and then divide by –3. Don't forget that when you divide or multiply by a negative number in an inequality you must flip the inequality sign. Thus, x < –19/3. Now plug in 3. We find, following the same steps, that when y=3, x < –6. Thus –7 is the correct answer.

This question is not as hard as it seems.  Remember that for real numbers, square roots cannot be taken of negative numbers.  Therefore, we know that this function is undefined for:

This is simple to solve.  Merely add  to both sides:

Then, divide by :

Therefore, quantity A is less than quantity B.  This means that quantity B is greater than it.

Recall that when you have an absolute value and an inequality like

,

this is the same as saying that  must be between  and .  You can rewrite it:

To solve this, you merely need to subtract  from all three values:

Since  is between  and , it could be both larger or smaller than .  Therefore, you cannot determine the relationship based on the given information.

For quantitative comparison questions involving a shared variable between quantities, the best approach is to test a positive integer, a negative integer, and a fraction. Half of our work is eliminated, however, because the question stipulates that x > 0.  We only need to check a positive integer and a positive fraction between 0 and 1. Plugging in 2, we see that quantity A is greater than quantity B. Checking 1/2, however, we find that quantity B is greater than quantity A. Thus the relationship cannot be determined.