# Algebra II : Setting Up Inequalities

## Example Questions

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### Example Question #1 : Inequalities

A grocer decides to mix cashews, almonds, and walnuts and sell the mixture for $14.99/lb. In order for the mixture to be profitable, he would like to make sure that the cost of making the product is less than the selling price. Cashews cost$8.99/lb

Almonds cost $6.99/lb Walnuts cost$10.99/lb

Which of the following inequality ensures that the grocer will make a profit by selling the mixture? For one pound of mixture, he uses the following quantities:

x: amount of cashews in the mix

y: amount of almonds in the mix

z: amount of walnuts in the mix

Explanation:

The grocer wants the cost of the mixture to be less than the selling price.

Selling price = 14.99

Find the cost of the mixture

1. cost of cashews in mix:

2. cost of almonds in mix:

3. cost of walnuts in mix:

The cost of the mixture =

Cost of the mixture is less than Selling price

### Example Question #2 : Inequalities

You're running a pineapple delivery company and you have  banked this month. You need to have  left over at the end of the month to ensure you can pay for your utilities and give your employees their salary however, each pineapple costs . How many pineapples can you order this month beforehand without running over your budget. Write an inequality to represent this situation.

Explanation:

If we write the number of pineapples ordered as , then  is the price of all the pineapples since each one costs . Now, it is alright if you have exactly  so it is an inclusive inequality, and you are drawing from an available amount of , so we can write this inequality in words as

6 dollars * (number of pineapples) from 1000 dollars must be greater than or equal to 400 dollars.

This can easily be converted to mathematical symbols as

### Example Question #3 : Inequalities

Angie and Kirsten are looking to buy a couch that is less than  . Kirsten has an individual spending limit of .

Which set of inequalities represents a solution for how much each girl will spend on the couch?

Explanation:

If Kirsten is represented by the variable , and Angie is represented by the variable , then the total amount Kirsten and Angie are spending can be represnted by the sum of . Since Angie and Kirsten only have  to spend on a couch,   must be less than . Therefore, . Kirsten only has  to spend on a couch. Therefore, .

### Example Question #4 : Inequalities

Write the following statements as a set of inequalities:

• The total number of shirts and pants I bought did not exceed 20.
• The cost of the total number of pants and shirts I bought was more than 50 dollars.

Where, s is the number of shirts, p is the number of pants and c, is the total cost.

Explanation:

The first statement says that the total number of pants and shirts does not exceed 20. This means that the total can be 20 but nothing more which is represented by:

The second sentence says that the cost was greater than 50 dollars which is represented by:

### Example Question #1 : Setting Up Inequalities

Set up the inequality:  Twice a number is less than twice the product of the number and five.

Explanation:

To set up this inequality, break up the statement into parts.  Assume  is the number.

Twice a number:

Is less than:

Product of the number and five:

Twice the product of the number and five:

Connect the statement.

### Example Question #1 : Setting Up Inequalities

Set up the inequality expressed in the word problem below:

Elise went to the grocery store, and bought a bunch of sweet potatoes and onions. She knows she has less than 12 altogether. What is the inequality that expresses how many sweet potatoes (x) and onions (y) she could have bought?

Explanation:

is the correct answer, because the only thing the problem tells us is that the sweet potatoes and onions add up to a number less than 12. It does not say if one is more than the other, and does not say it could equal 12. Therefore the less than sign should not be underlined.

### Example Question #1 : Inequalities

If  and , where is  located

The point will land on the x or y axis, therefore there is no quadrant for the scenario.

Explanation:

Because x is greater than zero, it will be to the right of the y axis. Because y is less than zero, it will be below the x axis. This is the fourth quadrant.

### Example Question #1 : Setting Up Inequalities

At a fair, there is a game where players step on a scale and weigh themselves. The objective of the game is for the host to guess the player's weight. A player loses if the host of the game can guess the player's weight within  pounds, inclusive. Suppose a player weighs  pounds. Write an inequality that represents the range of numbers such that the player loses. (Let  represent the guess weight.)

Explanation:

For the player to lose, the host has to guess within  pounds of the player's weight, inclusive. Thus, the host can guess any number between  pounds  and  pounds ; that is, if  is the weight the host guesses, then , which translates to

### Example Question #8 : Inequalities

Set up the following inequality:  Four less than three times a number squared is at least six.

Explanation:

Split up the inequality into parts.

A number squared:

Three times a number squared:

Four less than three times a number squared:

Is at least six:

Combine all the parts to form an inequality:

### Example Question #9 : Inequalities

Set up the inequality:  Four times the quantity of a number less than five is more than six.

Explanation:

Break up the sentence into parts.

A number less than five:

The quantity of a number less than five:

Four times the quantity of a number less than five:

More than six:

Combine the terms.