CP MathCP Math

Word Problems involving Word Problems involving inequalitiesinequalities

Quiz 6-4Quiz 6-4Graph the system of inequalities:Graph the system of inequalities:

11

2. What is a “solution” to a system of inequalities?2. What is a “solution” to a system of inequalities?

31

25.0 2

xy

xy

This is what “This is what “I’m sorryI’m sorry” ” looks like.looks like.

Inequality Word ProblemsInequality Word ProblemsNails cost $5 per pound. Screws cost $10 per pound. YouNails cost $5 per pound. Screws cost $10 per pound. You can spend up to $100.can spend up to $100. 1.1. What type of word problem is this? What type of word problem is this? (total income, total cost, profit, projectile motion, total number)(total income, total cost, profit, projectile motion, total number)

2.2. Let’s make the x-axis the number of pounds of Let’s make the x-axis the number of pounds of nails and the y-axis the number of pounds of nails and the y-axis the number of pounds of screws.screws.

What is a reasonable What is a reasonable domaindomain::What is a reasonable What is a reasonable rangerange??

x ≥ 0x ≥ 0nn ≥ 0 ≥ 0

ss ≥ 0 ≥ 0

Inequality Word ProblemsInequality Word ProblemsNails cost $5 per pound. Screws cost $10 per pound. YouNails cost $5 per pound. Screws cost $10 per pound. You can spend up to $100.can spend up to $100.

3. Write an inequality representing pounds of nails (use x-axis), 3. Write an inequality representing pounds of nails (use x-axis), and pounds of screws (use y-axis) you can buy.and pounds of screws (use y-axis) you can buy.

Can you buy Can you buy negativenegative pounds of nails? pounds of nails?

100 ≥ 5x + 10y100 ≥ 5x + 10y

x ≥ 0x ≥ 0

Can you buy Can you buy negativenegative pounds of screws? pounds of screws?

y ≥ 0y ≥ 0

100 ≥ 5100 ≥ 5nn + 10 + 10ss

nn ≥ 0 ≥ 0

ss ≥ 0 ≥ 0

Inequality Word ProblemsInequality Word ProblemsNails cost $5 per pound. Screws cost $10 per pound. YouNails cost $5 per pound. Screws cost $10 per pound. You can spend up to $100.can spend up to $100. 4. Graph the inequalities.4. Graph the inequalities.

100 ≥ 5x + 10y100 ≥ 5x + 10y

x ≥ 0x ≥ 0

AnyAny combination of pounds of screws combination of pounds of screws or nails in the shaded region will costor nails in the shaded region will cost between $0 and $100.between $0 and $100.

y ≥ 0y ≥ 0 1010

2020

Your turn:Your turn:1. 1. Adult tickets cost $7 and student tickets cost $5. The Adult tickets cost $7 and student tickets cost $5. The cost of producing the play cost $2500. cost of producing the play cost $2500.

a)a) Write an inequality that represents the Write an inequality that represents the number of student and adult tickets that need number of student and adult tickets that need to be sold so that the company doesn’t loose to be sold so that the company doesn’t loose any money.any money.

b)b) What is a reasonable What is a reasonable domaindomain??c)c) What is a reasonable What is a reasonable rangerange??

2. 2. Graph the system of inequalities to show all possibleGraph the system of inequalities to show all possible combinations of adult and student tickets needed tocombinations of adult and student tickets needed to ensure a profit. ensure a profit.

Your turn:Your turn:Apples cost $2 per pound. Peaches cost $3 per pound. YouApples cost $2 per pound. Peaches cost $3 per pound. You can spend up to $60.can spend up to $60. 3. 3. Write an inequality representing pounds of apples (use x-axis), Write an inequality representing pounds of apples (use x-axis), and pounds of peaches (use y-axis) you can buy.and pounds of peaches (use y-axis) you can buy.

60 ≥ 2a + 3p60 ≥ 2a + 3p

4. 4. Write two inequalities to show that you can’t buyWrite two inequalities to show that you can’t buy negative numbers of pounds of apples or peaches.negative numbers of pounds of apples or peaches.

a ≥ 0a ≥ 0 p ≥ 0p ≥ 0

5. 5. Graph the system of equations to show all possibleGraph the system of equations to show all possible combinations of the number of pounds of apples andcombinations of the number of pounds of apples and peaches you could buy.peaches you could buy.

Word ProblemsWord Problems You can work You can work at mostat most 30 hours. Your job as a waiter pays you $15 per hour (after 30 hours. Your job as a waiter pays you $15 per hour (after

taxes). Your job doing yard work pays you $10 per hour (after taxes). You need to earn taxes). Your job doing yard work pays you $10 per hour (after taxes). You need to earn

at least at least $150 per week to pay for gas, insurance, and your car payment. $150 per week to pay for gas, insurance, and your car payment.

What kind of problem type is this ?What kind of problem type is this ? Total incomeTotal income

Write the total income equation for this problem:Write the total income equation for this problem:

Total income = yard job pay + waiter job pay Total income = yard job pay + waiter job pay

Total number (of hours)Total number (of hours)

Write the total hours equation for this problem:Write the total hours equation for this problem:

Total hours = yard job hours + waiter job hours Total hours = yard job hours + waiter job hours

Word ProblemsWord Problems You can work You can work at mostat most 30 hours. Waiter pays $15 per hour Yard 30 hours. Waiter pays $15 per hour Yard

work pays $10 per hour. You need to earn work pays $10 per hour. You need to earn at least at least $150 per $150 per week. week.

Total income = yard job pay + waiter job pay Total income = yard job pay + waiter job pay

Total hours = yard job hours + waiter job hours Total hours = yard job hours + waiter job hours

What are the units of the x-axis?What are the units of the x-axis? Gardening hoursGardening hours

150 = 10x + 15y150 = 10x + 15y

30 = x + y30 = x + y

What are the units of the y-axis?What are the units of the y-axis? Waiter hoursWaiter hours

Word ProblemsWord Problems You can work You can work at mostat most 30 hours. Waiter pays $15 per hour Yard 30 hours. Waiter pays $15 per hour Yard

work pays $10 per hour. You need to earn work pays $10 per hour. You need to earn at least at least $150 per $150 per week. week.

Total income = yard job pay + waiter job pay Total income = yard job pay + waiter job pay

Total hours = yard job hours + waiter job hours Total hours = yard job hours + waiter job hours 150 = 10x + 15y150 = 10x + 15y

30 = x + y30 = x + y

Write a system of inequalities to represent the situation.Write a system of inequalities to represent the situation.

30 ≥ x + y30 ≥ x + y

150 ≤ 10x + 15y150 ≤ 10x + 15y x ≥ 0x ≥ 0

y ≥ 0y ≥ 0

x = gardening hoursx = gardening hours y = waiter hoursy = waiter hours

Graph the system of Graph the system of inequalities:inequalities:

x ≥ 0x ≥ 0

y ≥ 0y ≥ 0 1010

1515

30 ≥ x + y30 ≥ x + y

150 ≤ 10x + 15y150 ≤ 10x + 15y

x = gardening hoursx = gardening hours y = waiter hoursy = waiter hours

3030

3030

what part is shaded?what part is shaded?

You can work You can work at mostat most 20 hours per week. Your job as a 20 hours per week. Your job as a

bagger at Harmon’s pays you $5 per hour (after taxes). Your job as a dog walker pays you $7 bagger at Harmon’s pays you $5 per hour (after taxes). Your job as a dog walker pays you $7 per hour (after taxes. You need to earn at least $70 per week to pay for gas, insurance, and per hour (after taxes. You need to earn at least $70 per week to pay for gas, insurance, and your car payment. your car payment.

6. 6. Write a system of inequalities to represent the situation.Write a system of inequalities to represent the situation.

Your turn:Your turn:

7. 7. Graph the system of equations to show the possible Graph the system of equations to show the possible combinations of hours you can work at Harmon’s and as a combinations of hours you can work at Harmon’s and as a dog walker.dog walker.

Quadratic Inequality Word ProblemsQuadratic Inequality Word Problems

A ball is dropped from the roof of a A ball is dropped from the roof of a 500 foot 500 foot building. The building. The height (in feet) as a function of time (in seconds) of the height (in feet) as a function of time (in seconds) of the ball is given by the following equation: ball is given by the following equation:

For what period of time is the ball above 400 feet?For what period of time is the ball above 400 feet?

50016)( 2 tts

400400

Quadratic Inequality Word ProblemsQuadratic Inequality Word Problems

A ball is dropped from the roof of a A ball is dropped from the roof of a 500 foot 500 foot building. building. For what period of time is the ball above 400 feet?For what period of time is the ball above 400 feet?

400400

Which option Which option do you use to do you use to find the time find the time where the two where the two graphs cross?graphs cross?

Quadratic Inequality Word ProblemsQuadratic Inequality Word Problems

A ball is dropped from the roof of a A ball is dropped from the roof of a 500 foot 500 foot building. building. For what period of time is the ball above 400 feet?For what period of time is the ball above 400 feet?

400400

2.5 sec2.5 sec

From 0 secondsFrom 0 seconds to just less than to just less than 2.5 seconds. 2.5 seconds.

5.20 t

Your turn:Your turn:A ball is dropped from the roof of a A ball is dropped from the roof of a 500 foot 500 foot building. The building. The

height of the ball as a function of time is given by: height of the ball as a function of time is given by:

7.7. For what period of time will the ball be above 200 feet For what period of time will the ball be above 200 feet above ground level?above ground level?

8. 8. For what period of time will the ball be between 100 and For what period of time will the ball be between 100 and 300 feet?300 feet?

50016)( 2 tts

Your turn:Your turn:An instrumented package that measures air pressure, An instrumented package that measures air pressure,

temperature, and humidity is launched upward using large temperature, and humidity is launched upward using large rubber bands obtained from a medical supply store. The rubber bands obtained from a medical supply store. The height of the package above ground as a function of time is height of the package above ground as a function of time is given by the equation above. given by the equation above.

9. 9. For what period of time will the package be above 700 feet?For what period of time will the package be above 700 feet?

10. 10. For what period of time will the package be below 300 For what period of time will the package be below 300 feet above ground level?feet above ground level?

ttts 20016)( 2

The end.The end.

Inequality Word ProblemsInequality Word Problems

y ≤ 0.75xy ≤ 0.75x

No more thanNo more than 75% of the original cost. 75% of the original cost.

A clothing store is having a sale. They say thatA clothing store is having a sale. They say that “ “all hoodies are all hoodies are at least 25% offat least 25% off the regular price”. the regular price”. 1.1. Write an inequality representing this saleWrite an inequality representing this sale (use (use x for regular price x for regular price and and y for sale pricey for sale price

Think about itThink about it: Regular price $10, : Regular price $10, at least 25% offat least 25% off means the means the priceprice will be what? will be what?

What does “sale cost will be What does “sale cost will be no more thanno more than 75% of the 75% of the original price mean? original price mean?

Inequality Word ProblemsInequality Word Problems

y ≤ 0.75xy ≤ 0.75x

A clothing store is having a sale. They say thatA clothing store is having a sale. They say that “ “all hoodies are all hoodies are at least 25% offat least 25% off the regular price”. the regular price”.

““Sale cost will be Sale cost will be no more thanno more than 75% of the 75% of the original price.” original price.”

Can the Can the sale price sale price be a negative number?be a negative number?

What is the smallest price they could charge?What is the smallest price they could charge?

0 ≤ y ≤ 0.75x0 ≤ y ≤ 0.75x

Can the Can the regular price regular price be a negative number?be a negative number?

x ≥ 0x ≥ 0

Inequality Word ProblemsInequality Word ProblemsA clothing store is having a sale. They say thatA clothing store is having a sale. They say that “ “all hoodies are all hoodies are at least 25% offat least 25% off the regular price”. the regular price”.

0 ≤ y ≤ 0.75x0 ≤ y ≤ 0.75x

Compound inequality: y ≥ 0 and y ≤ 0.75xCompound inequality: y ≥ 0 and y ≤ 0.75x

2. Graph the inequalities on the same graph2. Graph the inequalities on the same graph (use (use x for regular price x for regular price and and y for sale pricey for sale price

x ≥ 0x ≥ 0

regular priceregular price

salesale priceprice

Inequality Word ProblemsInequality Word ProblemsA clothing store is having a sale. They say thatA clothing store is having a sale. They say that “ “all hoodies are all hoodies are at least 25% offat least 25% off the regular price”. the regular price”.

0 ≤ y ≤ 0.75x0 ≤ y ≤ 0.75x

y ≥ 0 and y ≤ 0.75xy ≥ 0 and y ≤ 0.75x

2. Graph the inequalities on the same graph2. Graph the inequalities on the same graph (use (use x for regular price x for regular price and and y for sale pricey for sale price

x ≥ 0x ≥ 0

regular priceregular price

salesale priceprice

100100

7575

If x = 100 If x = 100 y = ? y = ?