# PSAT Math : Inequalities

## Example Questions

### Example Question #21 : Inequalities

What is the greatest value of  that makes

a true statement?

Explanation:

Find the solution set of the three-part inequality as follows:

The greatest possible value of  is the upper bound of the solution set, which is 277.

### Example Question #22 : Inequalities

What is the least value of  that makes

a true statement?

Explanation:

Find the solution set of the three-part inequality as follows:

The least possible value of  is the lower bound of the solution set, which is 139.

### Example Question #23 : Inequalities

Give the solution set of the inequality:

None of the other responses gives the correct answer.

Explanation:

Divide each of the three expressions by , or, equivalently, multiply each by its reciprocal, :

or, in interval form,

.

### Example Question #24 : Inequalities

Give the solution set of the following inequality:

None of the other responses gives the correct answer.

Explanation:

or, in interval notation, .

### Example Question #25 : Inequalities

Which of the following numbers could be a solution to the inequality ?

Explanation:

In order for a negative multiple to be greater than a number and a positive multiple to be less than that number, that number must be negative itself. -4 is the only negative number available, and thus the correct answer.

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

Each of the following is equivalent to

xy/z * (5(x + y))  EXCEPT:

5x² + y²/z

5x²y + 5xy²/z

xy(5x + 5y)/z

xy(5y + 5x)/z

5x² + y²/z

Explanation:

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.

### Example Question #2 : How To Find The Solution To An Inequality With Division

Let S be the set of numbers that contains all of values of x such that 2x + 4 < 8. Let T contain all of the values of x such that -2x +3 < 8. What is the sum of all of the integer values that belong to the intersection of S and T?

-7

-3

-2

2

0

-2

Explanation:

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.

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

The Spanish club wants to make and sell some pizzas for a fundraiser. It will cost $300 to rent the equipment to make the pizzas and$2 worth of ingredients to make each pizza. The pizzas will be sold for $5.50 apiece. How many pizzas must be made and sold for the Spanish club to make a profit of at least$600?

Explanation:

Let  be the number of pizzas made and sold. Each pizza will require $2 worth of ingredients, so the ingredients in total will cost . Add this to the cost to rent the equipment and the cost will be . The pizzas will cost$5.50 each, so the money raised will be .

The profit will be the difference between the revenue and the cost -

The Spanish club wants a profit of at least \$600, so we set up and solve the inequality:

The Spanish club must sell at least 258 pizzas to earn a profit.

### Example Question #2 : How To Find The Solution To An Inequality With Division

Solve the inequality.

Explanation:

* Notice that when we multiply or divide both sides by a negative number the ineqaulity sign changes orientation.

Solve for :