### All GRE Math Resources

## Example Questions

### Example Question #23 : Inequalities

Quantitative Comparison

Quantity A: 3*x* + 4*y*

Quantity B: 4*x* + 3*y*

**Possible Answers:**

The two quantities are equal.

Quantity B is greater.

Quantity A is greater.

The relationship cannot be determined from the information given.

**Correct answer:**

The relationship cannot be determined from the information given.

The question does not give us any specifics about the variables *x* and *y*.

If we substitute the same numbers for x and y (say, x = 1 and y = 1), the two expressions are equal.

If we substitute different number in for x and y (say, x = 2 and y = 1), the two expressions are not equal.

If there are two possible outcomes, then we need more information to determine which quantity is greater. Don't be afraid to pick "The relationship cannot be determined from the information given" as an answer choice on the GRE!

### Example Question #1 : How To Find The Solution To An Inequality With Addition

Let be an integer such that .

Quantity A:

Quantity B:

**Possible Answers:**

The relationship cannot be determined from the information given.

Quantity A and Quantity B are equal.

Quantity B is greater.

Quantity A is greater.

**Correct answer:**

Quantity A and Quantity B are equal.

The expression can be rewritten as .

The only integer that satisfies the inequality is 0.

Thus, Quantity A and Quantity B are equal.

### Example Question #25 : Inequalities

Find all solutions of the inequality .

**Possible Answers:**

All .

All .

All .

All .

All ..

**Correct answer:**

All ..

Start by subtracting 3 from each side of the inequality. That gives us . Divide both sides by 2 to get . Therefore every value for where is a solution to the original inequality.

### Example Question #26 : Inequalities

Find all solutions of the inequality .

**Possible Answers:**

All .

All .

All .

All .

All .

**Correct answer:**

All .

Start by subtracting 13 from each side. This gives us . Then subtract from each side. This gives us . Divide both sides by 2 to get . Therefore all values of where will satisfy the original inequality.

### Example Question #1 : Inequalities

What values of *x* make the following statement true?

|*x* – 3| < 9

**Possible Answers:**

6 < *x* < 12

*x* < 12

–12 < *x* < 6

–6 < *x* < 12

–3 < *x* < 9

**Correct answer:**

–6 < *x* < 12

Solve the inequality by adding 3 to both sides to get *x* < 12. Since it is absolute value, *x* – 3 > –9 must also be solved by adding 3 to both sides so: *x* > –6 so combined.

### Example Question #1 : How To Find The Solution To An Inequality With Addition

If –1 < *w* < 1, all of the following must also be greater than –1 and less than 1 EXCEPT for which choice?

**Possible Answers:**

3*w*/2

|*w*|^{0.5}

*w*^{2}

|*w*|

*w*/2

**Correct answer:**

3*w*/2

3*w*/2 will become greater than 1 as soon as *w* is greater than two thirds. It will likewise become less than –1 as soon as *w* is less than negative two thirds. All the other options always return values between –1 and 1.

### Example Question #1 : How To Find The Solution To An Inequality With Addition

Solve for .

**Possible Answers:**

**Correct answer:**

Absolute value problems always have two sides: one positive and one negative.

First, take the problem as is and drop the absolute value signs for the positive side: z – 3 ≥ 5. When the original inequality is multiplied by –1 we get z – 3 ≤ –5.

Solve each inequality separately to get z ≤ –2 or z ≥ 8 (the inequality sign flips when multiplying or dividing by a negative number).

We can verify the solution by substituting in 0 for z to see if we get a true or false statement. Since –3 ≥ 5 is always false we know we want the two outside inequalities, rather than their intersection.

### Example Question #1 : How To Find The Solution To An Inequality With Addition

If and , then what is the value of ?

**Possible Answers:**

**Correct answer:**

To solve this problem, add the two equations together:

The only answer choice that satisfies this equation is 0, because 0 is less than 4.

### Example Question #1 : How To Find The Solution To An Inequality With Addition

What values of make the statement true?

**Possible Answers:**

**Correct answer:**

First, solve the inequality :

Since we are dealing with absolute value, must also be true; therefore:

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