### All GMAT Math Resources

## Example Questions

### Example Question #1 : Solving Inequalities

How many integers can complete this inequality?

**Possible Answers:**

**Correct answer:**

3 is added to each side to isolate the term:

Then each side is divided by 2 to find the range of :

The only integers that are between 5 and 9 are 6, 7, and 8.

The answer is 3 integers.

### Example Question #2 : Solving Inequalities

Solve .

**Possible Answers:**

**Correct answer:**

Subtract 10:

Divide by 3:

We must carefully check the endpoints. is greater than and cannot equal , yet CAN equal 2. Therefore should have a parentheses around it, and 2 should have a bracket: is in

### Example Question #3 : Solving Inequalities

Solve .

**Possible Answers:**

**Correct answer:**

Subtract 3 from both sides:

Divide both sides by :

Remember: when dividing by a negative number, reverse the inequality sign!

Now we need to decide if our numbers should have parentheses or brackets. is strictly greater than , so should have a parentheses around it. Since there is no upper limit here, is in .

Note: Infinity should ALWAYS have a parentheses around it, NEVER a bracket.

### Example Question #4 : Solving Inequalities

Solve .

**Possible Answers:**

**Correct answer:**

must be positive, except when . When , .

Then we know that the inequality is only satisfied when , and . Therefore , which in interval notation is .

Note: Infinity must always have parentheses, not brackets. has a parentheses around it instead of a bracket because is less than , not less than or equal to .

### Example Question #5 : Solving Inequalities

Solve .

**Possible Answers:**

**Correct answer:**

The roots we need to look at are

:

Try

, so

does not satisfy the inequality.

:

Try

so does satisfy the inequality.

:

Try

so does not satisfy the inequality.

:

Try .

so satisfies the inequality.

Therefore the answer is and .

### Example Question #6 : Solving Inequalities

Find the domain of .

**Possible Answers:**

all positive real numbers

all non-negative real numbers

all real numbers

**Correct answer:**

We want to see what values of x satisfy the equation. is under a radical, so it must be positive.

### Example Question #7 : Solving Inequalities

Solve the inequality:

**Possible Answers:**

**Correct answer:**

When multiplying or dividing by a negative number on both sides of an inequality, the direction of the inequality changes.

### Example Question #8 : Solving Inequalities

Find the solution set for :

**Possible Answers:**

**Correct answer:**

Subtract 7:

Divide by -1. Don't forget to switch the direction of the inequality signs since we're dividing by a negative number:

Simplify:

or in interval form, .

### Example Question #9 : Solving Inequalities

Which of the following is equivalent to ?

**Possible Answers:**

**Correct answer:**

To solve this problem we need to isolate our variable .

We do this by subtracting from both sides and subtracting from both sides as follows:

Now by dividing by 3 we get our solution.

or

### Example Question #10 : Solving Inequalities

How many integers satisfy the following inequality:

**Possible Answers:**

Three

Four

One

Two

Five

**Correct answer:**

Two

There are two integers between 2.25 and 4.5, which are 3 and 4.

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