Inequalities - GRE Quantitative Reasoning

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

50xgeq 1200+20x

xgeq 40

The answer is:

y-7< x+2y-14

-y-7< x-14

-1(-y< x-7)

y > -x+7

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

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.

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.

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

50xgeq 1200+20x

xgeq 40

The answer is:

y-7< x+2y-14

-y-7< x-14

-1(-y< x-7)

y > -x+7

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

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.

The equation simplifies to x > -1.41. -1 is the answer.

3x+7 geq -2x+4

3x geq -2x-3

5x geq -3

xgeq -$\frac{3}{5}$

$(x-3)circ$\frac{x^2$-9}{x+3}$

Let us simplify the second expression. We know that:

$(x^2$-9)=(x+3)(x-3)

So we can cancel out as follows:

$$\frac{x^2$-9}{x+3}$=\frac{(x+3)(x-3)}{(x+3)}$=x-3

$(x-3)=\frac{x^2$-9}{x+3}$

Start by simplifying the expression by distributing through the parentheses to .

Subtract from both sides to get .

Next subtract 9 from both sides to get . Then divide by 4 to get which is the same as .

Start by simplifying each side of the inequality by distributing through the parentheses.

This gives us .

Add 6 to both sides to get .

Add to both sides to get .

Divide both sides by 13 to get .

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.