Problem-Solving Questions - GMAT Quantitative

By definition:

If , then ,and subsequently,

If , then ,and subsequently,

To solve a quadratic inequality, move all expressions to the left first:

The square of a real number cannot be less than 0, so

, the only solution.

The square of a real number must be nonnegative, so this is a true statement regardless of the value of . The solution set is the set of all real numbers

To solve a quadratic inequality, move all expressions to the left first

The boundary points of the solution set will be the points at which:

; that is, ; or,

; that is,

These values will be included in the solution set, since equality is allowed by the inequality symbol.

Test the intervals

by choosing a value in each interval and testing the truth of the inequality.

: test

False - exclude this interval

: test

True - include this interval

: test

False - exclude this interval

is the solution set.

The total number of balls in the box will be

.

Since

,

it follows that the number of balls is

.

The frequencies out of 55 of each outcome from 1 to 10, in order, is as follows:

Their respective probabilities are their frequencies divided by 55:

The expected value of the payment that the church will have to make per game can be calculated by multiplying the frequency of each outcome by the respective payment:

...and then adding these products.

The carnival should expect to pay a prize of $4 per draw, so to expect an average profit of $1 per game, it should charge $5 per play.

Let's look at and see if any of them are functions.

1. = {(2, 3), (1, 4), (2, 1), (3, 2), (4, 4)}: This cannot be a function of because two of the ordered pairs, (2, 3) and (2, 1) have the same number (2) as the first coordinate.

2. = {(3, 1), (4, 2), (1, 1)}: This cannot be a function of because it contains no ordered pair with first coordinate 2. Because the set = {1, 2, 3, 4}, we need an ordered pair of the form (2, ) .

3. = {(2, 1), (3, 4), (1, 4), (2, 1), (4, 4)}: This is a function. Even though two of the ordered pairs have the same number (2) as the first coordinate, is still a function of because (2, 1) is simply repeated twice, so the two ordered pairs with first coordinate 2 are equal.

Write this as a compound equation and solve each separately.

This gives us three possible solutions - . We check all three.

This is a false statement so we can eliminate as a false "solution".

2 is a solution.

3 is a solution.

The solution set is .

If , then and are both positive, so

If , then , then is positive and is negative, so

If , then and are both negative, so

We are looking for the union of the sets. That means we want the elements of A OR B.

So = {2, 3, 4, 5, 7}.

Order and repetition do NOT change a set. Therefore, the set we want to describe contains the numbers 1, 3, and 4. The only set that doesn't contain all 3 of these numbers is , so it is the set that does not equal the rest of the sets.

Add each element of to each element of .

= {1 + 3, 1 + 4, 1 + 6, 4 + 3, 4 + 4, 4 + 6} = {4, 5, 7, 8, 10}

If , then and are both positive, so

If , then , then is positive and is negative, so

If , then and are both negative, so

If , then ,and subsequently,

If , then ,and subsequently,

Factor the expression:

Rewrite:

or

We are looking for the union of the sets. That means we want the elements of A OR B.

So = {2, 3, 4, 5, 7}.

To solve a quadratic inequality, move all expressions to the left first:

The square of a real number cannot be less than 0, so

, the only solution.

The square of a real number must be nonnegative, so this is a true statement regardless of the value of . The solution set is the set of all real numbers

To solve a quadratic inequality, move all expressions to the left first

The boundary points of the solution set will be the points at which:

; that is, ; or,

; that is,

These values will be included in the solution set, since equality is allowed by the inequality symbol.

Test the intervals

by choosing a value in each interval and testing the truth of the inequality.

: test

False - exclude this interval

: test

True - include this interval

: test

False - exclude this interval

is the solution set.

Let's look at and see if any of them are functions.

1. = {(2, 3), (1, 4), (2, 1), (3, 2), (4, 4)}: This cannot be a function of because two of the ordered pairs, (2, 3) and (2, 1) have the same number (2) as the first coordinate.

2. = {(3, 1), (4, 2), (1, 1)}: This cannot be a function of because it contains no ordered pair with first coordinate 2. Because the set = {1, 2, 3, 4}, we need an ordered pair of the form (2, ) .

3. = {(2, 1), (3, 4), (1, 4), (2, 1), (4, 4)}: This is a function. Even though two of the ordered pairs have the same number (2) as the first coordinate, is still a function of because (2, 1) is simply repeated twice, so the two ordered pairs with first coordinate 2 are equal.

We are looking for the union of the sets. That means we want the elements of A OR B.

So = {2, 3, 4, 5, 7}.

To solve a quadratic inequality, move all expressions to the left first:

The square of a real number cannot be less than 0, so

, the only solution.