Introduction to Linear Inequality Word Problems
Linear inequality word problems are solved much the same way as linear equality word problems. There are two important differences. Multiplying and dividing both sides of an inequality by a negative quantity requires that the sign reverse. You must also decide which inequality sign to use: <, >, ≤, and ≥. The following tables should help.
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A < B
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A > B
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A is less than B
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A is greater than B
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A is smaller than B
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A is larger than B
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B is greater than A
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B is less than A
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B is larger than A
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A is more than B
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A ≤ B
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B ≤ A
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A is less than or equal to B
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B is less than or equal to A
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A is not more than B
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B is not more than A
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B is at least A
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A is at least B
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B is A or more
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A is B or more
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A is no greater than B
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B is no greater than A
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B is no less than A
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A is no less than B
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Some word problems give two alternatives and ask for what interval of the variable is one alternative more attractive than the other. If the alternative is between two costs, for example, in order for the cost of A to be more attractive than the cost of B, solve “Cost of A < Cost of B.” If the cost of A to be no more than the cost of B (also the cost of A to be at least as attractive as the cost of B), solve “Cost of A ≤ Cost of B.” If the alternative is between two incomes of some kind, for the income of A to be more attractive than the income of B, solve “Income of A > Income of B.” If the income of A is to be at least as attractive as the income of B (also the income of A to be no less attractive than the income of B), solve “Income of A ≥ Income of B.”
Linear Inequalities for Business Word Problems
Some of the following examples and practice problems are business problems. Let us review a few business formulas. Revenue is normally the price per unit times the number of units sold. For instance if an item sells for $3.25 each and x represents the number of units sold, the revenue is represented by 3.25 x (dollars). Cost tends to consist of overhead costs (sometimes called fixed costs ) and production costs (sometimes called variable costs ). The overhead costs will be a fixed number (no variable). The production costs is usually computed as the cost per unit times the number of units sold. The total cost is usually the overhead costs plus the production costs. Profit is revenue minus cost. If a problem asks how many units must be sold to make a profit, solve “Revenue > Total Cost.”
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?
Solutions
- Let x = amount invested in the corporate bond
500,000 – x = amount invested in the treasury bond
0.08 x = annual interest from the corporate bond
0.0525(500,000 – x ) = annual interest from the treasury bond
Corporate bond interest + Treasury bond interest ≥ 30,000
The administrator should invest at least $136,363.64 in the corporate bond in order to receive at least $30,000 per year in interest payments.
- Let x = number of corn dogs sold per day
2 x = revenue
200 + 0.35 x = overhead costs + production costs = total cost
Kelly needs to sell at least 400 corn dogs in order for her daily profit to be at least $460.
- Let x = number of snow cones sold per month
1.50 x = revenue
600 + 0.25 x = overhead costs + production costs = total cost
Revenue > Cost
The owner should sell more than 480 snow cones per month to make a profit.
- Let x = number of tee shirts sold per month
18 x = revenue
1500 + 8 x = overhead costs + production costs = total cost
18 x –(1500 + 8 x )= profit
Profit ≥ 3500
At least 500 tee shirts would need to be sold each month to make a monthly profit of at least $3500.
- Let x = number of daily miles
The $24 option costs 24 + 0.80 x per day.
The most a customer could drive is 200 miles per day in order for the $24 plan to cost no more than the $40 plan.
- The first 50 hours are free under the $12 plan, so let x represent the number of hours used beyond 50 hours. Each hour beyond 50 costs 0.65 x .
A family would need to use more than 20 hours per month beyond 50 hours (or more than 70 hours per month) in order for the unlimited plan to cost less than the limited hour plan.
- Let x = number of times Sharon uses her skates The cost to rent skates is 3 x .
Sharon would need to use her skates more than 20 times to justify purchasing them instead of renting them.
- Let x = number of kilowatt-hours used per month
28 + 0.07 x = monthly bill
The James family can use no more than 2600 kilowatt-hours per month in order to keep their electricity costs within their budget.
- Let x = amount spent at the store annually 0.05 x = extra 5% purchase charge per year
A shopper would need to spend at least $800 per year to justify the $40 annual fee.
- Let x = annual sales level
0.08 x = annual commission
15,000 + 0.08 x = annual salary plus commission
The sales clerk would need an annual sales level of $125,000 or more in order for the salary plus commission option to be at least as attractive as the straight salary option.
Practice problems for this concept can be found at: Algebra Linear Inequalities Practice Test.
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