**Examples**

**Example 1:**

A manufacturing plant, which produces compact disks, has monthly overhead costs of $6000. Each disk costs 18 cents to produce and sells for 30 cents. How many disks must be sold in order for the plant to make a profit?

Let *x* = number of CDs produced and sold monthly

Cost = 6000 + 0.18 *x* and Revenue = 0.30 *x*

Revenue > Cost

The plant should produce and sell more than 50,000 CDs per month in order to make a profit.

**Example 2: **

Mary inherited $16,000 and will deposit it into two accounts, one paying interest and the other paying interest. What is the most she can deposit into the account so that her interest at the end of a year will be at least $960?

Let *x* = amount deposited in the account

0.055 *x* = interest earned at

16,000 – *x* = amount deposited in the account

0.0675(16,000 – *x* ) = interest earned at

Interest earned at + Interest earned at

Mary can invest no more than $9600 in the account in order to receive at least $960 interest at the end of the year.

**Example 3:**

An excavating company can rent a piece of equipment for $45,000 per year. The company could purchase the equipment for monthly costs of $2500 plus $20 for each hour it is used How many hours per year must the equipment be used to justify purchasing it rather than renting it?

Let *x* = number of hours per year the equipment is used

The monthly payments amount to 12(2500) = 30,000 dollars annually. The annual purchase cost is 30,000 + 20 *x* .

The equipment should be used less than 750 hours annually to justify purchasing it rather than renting it.

**Example 4:**

An amusement park sells an unlimited season pass for $240. A daily ticket sells for $36. How many times would a customer need to use the ticket in order for the season ticket to cost less than purchasing daily tickets?

Let *x* = number of daily tickets purchased per season

36 *x* = daily ticket cost

Season ticket cost < daily ticket cost

A customer would need to use the ticket more than times (or 7 or more times) in order for the season ticket to cost less than purchasing daily tickets.

**Example 5:**

Bank A offers a certificate of deposit and Bank B offers a certificate of deposit but will give a $25 bonus at the end of the year. What is the least amount a customer would need to deposit at Bank A to make Bank A’s offer no less attractive than Bank B’s offer?

Let *x* = amount to deposit

If *x* dollars is deposited at Bank A, the interest at the end of the year would be 0.065 *x* . If *x* dollars is deposited at Bank B, the interest at the end of the year would be 0.0575 *x* . The total income from Bank B would be 25 + 0.0575 *x* .

Income from Bank A ≥ Income from Bank B

A customer would need to deposit at least $3333.33 in Bank A to earn no less than would be earned at Bank B.

**Linear Inequality Word Problems Practice Problems **

**Practice**

- A scholarship administrator is using a $500,000 endowment to purchase two bonds. A corporate bond pays 8% interest per year and a safer treasury bond pays interest per year. If he needs at least $30,000 annual interest payments, what is the least he can spend on the corporate bond?
- Kelly sells corn dogs at a state fair. Booth rental and equipment rental total $200 per day. Each corn dog costs 35 cents to make and sells for $2. How many corn dogs should she sell in order to have a daily profit of at least $460?
- The owner of a snow cone stand pays $200 per month to rent his equipment and $400 per month for a stall in a flea market. Each snow cone costs 25 cents to make and sells for $1.50. How many snow cones does he need to sell in order to make a profit?
- A tee-shirt stand can sell a certain sports tee shirt for $18. Each shirt costs $8 in materials and labor. Monthly fixed costs are $1500. How many tee shirts must be sold to guarantee a monthly profit of at least $3500?
- A car rental company rents a certain car for $40 per day with unlimited mileage or $24 per day plus 80 cents per mile. What is the most a customer can drive the car per day for the $24 option to cost no more than the unlimited mileage option?
- An internet service provider offers two plans. One plan costs $25 per month and allows unlimited internet access. The other plan costs $12 per month and allows 50 free hours plus 65 cents for each additional hour. How many hours per month would a customer need to use in order for the unlimited access plan be less expensive than the other plan?
- Sharon can purchase a pair of ice skates for $60. It costs her $3 to rent a pair each time she goes to the rink. How many times would she need to use the skates to make purchasing them more attractive than renting them?
- The James family has $210 budgeted each month for electricity. They have a monthly base charge of $28 plus 7 cents per kilowatt-hour. How many kilowatt-hours can they use each month to stay within their budget?
- A warehouse store charges an annual fee of $40 to shop there. A shopper without paying this fee can still shop there if he pays a 5% buyer’s premium on his purchases. How much would a shopper need to spend at the store to make paying the annual $40 fee at least as attractive as paying the 5% buyer’s premium?
- A sales clerk at an electronics store is given the option for her salary to be changed from a straight annual salary of $25,000 to an annual base salary of $15,000 plus an 8% commission on sales. What would her annual sales level need to be in order for this option to be at least as attractive as the straight salary option?

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