Solving Inequalities

Solving Inequalities

Module 3 Lesson 14

The annual County Carnival is being held this summer and will last 5.5 days.

• The roller coaster costs $6 to ride. 

• The operator of the ride must pay $200 per day for the ride rental and $65 per day for a safety inspection. 

• You want to make a profit of at least $1,000 each day

• Write an inequality that can be used to find the minimum number of people, 𝑝, which must ride the roller coaster each day to make the daily profit. 

Multiple Choice

Which inequality should you use?

• $6p\ -\ 265\ \ge\ 1000$

• $6p\ -265\ \le\ 1000$

• $6p\ +\ 265\ \ge\ 1000$

• $6p\ +\ 265\ \le\ 1000$

Write and Solve the Inequality

Multiple Choice

What is the minimum number of people that must ride the roller coaster?

• 210

• 211

• 212

• 210.83

Why is the answer 211?

• You won't have enough with 210

• You can't sell a fraction of a ticket 

A youth summer camp has budgeted $2,000 to attend the carnival.

• The cost for each camper is $17.95, which includes general admission and two meals.

• Expenses include $250 for chaperones and $350 for transportation

• What is the greatest number of campers who can attend the carnival if the camp must stay within its budgeted amount?

Multiple Choice

Which inequality should be used to answer this question?  What is the greatest number of campers who can attend the carnival if the camp must stay within its budgeted amount?

• 17.95c+600<2000

•  $17.95c+600>2000$  

•  $17.95p+600\ge2000$  

•  $17.95p\ +\ 600\ \le\ 2000$  

Write and solve this inequality

Multiple Choice

What is the greatest number of campers who can attend the carnival?

• 76

• 77

• 78

• 79

• 80

The carnival owner pays the owner of an exotic animal exhibit $650 the exhibit.

• If the owner of the animal exhibit wants to make more than $500 in profits for the 5 ½ days.

• What is the greatest daily insurance rate he can afford to pay?

• Write an inequality

Multiple Choice

Which is the correct inequality?

• 5.5i +650>500

• 5.5i+650<500

• -5.5i+650>500

• -5.5i+650<500

Multiple Choice

What is the solution?

• i < 27.27

• i > 27.27

• $i\ \le\ 27.27$

• $i\ \ge\ 27.27$

Multiple Choice

Was that inequality symbol preserved or reversed?

• preserved

• reversed

Multiple Select

Why was the inequality symbol reversed?

• Adding a negative

• Subtracting a negative

• Multiplying a negative

• Dividing a negative

The only times we reverse an inequality symbol are when we

• multiply by a negative across the inequality symbol

• divide by a negative across the inequality symbol

Example 3

Several vendors at the carnival sell products and advertise their businesses. Shane works for a recreational company that sells ATVs, dirt bikes, snowmobiles, and motorcycles. His boss paid him $500 for working all of the days at the carnival plus 5% commission on all of the sales made at the carnival. 

What was the minimum amount of sales Shane needed to make if he earned more than $1,500?

Multiple Choice

Which inequality should be used to find the solution?

• 0.005s+500<1500

• 0.05s+500>1500

• 0.05s-500<1500

• 0.05s-500>1500

Write and solve this inequality

Multiple Choice

What is the solution?

• s > 2000

• s < 2000

• s > 20,000

• s < 20,000

Lesson Summary 

• To solve inequalities we use if-then moves to make 0’s and 1’s

•  Adding or subtracting opposites will make 0’s

• Multiplying and dividing numbers makes 1’s

Exit Ticket 

Games at the carnival cost $3 each. The prizes awarded to winners cost $145.65. How many games must be played to make at least $50?

Write and solve

 $3g-145.65\ge50$  

Multiple Choice

Which inequality should be used to find the solution?

• 3g-145.65>50

• 3g+145.65>50

• $3g-145.65\ge50$

• $3g+145.65\ge50$

Multiple Choice

The solution

• 64

• 65

• 66

• 67

Problem Set Module 3 Lesson 14

As a salesperson, Jonathan is paid $𝟓𝟎 per week plus 𝟑% of the total amount he sells

• This week, he wants to earn at least $𝟏𝟎𝟎.

• Write an inequality with integer coefficients for the total sales needed to earn at least $𝟏𝟎0

• describe what the solution represents

Multiple Choice

Which inequality can Jonathan use to find how much he needs to sell?

• 0.03p+50>100

• 0.03p-50>100

• $0.03p+50\ge100$

• $0.03p-50\ge100$

Blood Pressure

Systolic blood pressure is the higher number in a blood pressure reading. It is measured as the heart muscle contracts. Heather was with her grandfather when he had his blood pressure checked. The nurse told him that the upper limit of his systolic blood pressure is equal to half his age increased by 𝟏𝟏𝟎.

Heather’s grandfather is 𝟕𝟔 years old. 

Write and solve this inequality

Multiple Choice

Which inequality should Heather use?

• $p\le\frac{1}{2}a+110$

• $p\ge\frac{1}{2}a+110$

• $p=\frac{1}{2}a+110$

Multiple Choice

What is normal for his systolic blood pressure?

• $p\le148$

• $p\ge148$

• $p=148$

Marathon Dance

Traci collects donations for a dance marathon. One group of sponsors will donate a total of $𝟔 for each hour she dances. Another group of sponsors will donate $𝟕𝟓 no matter how long she dances. What number of hours, to the nearest minute, should Traci dance if she wants to raise at least $𝟏, 𝟎𝟎𝟎? 

Write and solve the inequality

Multiple Choice

Which inequality should you use?

• $6h+75\ge1000$

• $6h-75\ge1000$
  

Multiple Choice

The solution

• $h\ge154$

• $h\ge154\frac{1}{6}$

• $h\ge155$

 Jack’s age is three years more than twice the age of his younger brother, Jimmy. If the sum of their ages is at most 𝟏𝟖, find the greatest age that Jimmy could be.

Write and solve the inequality

Multiple Choice

Which inequality should you use?

• $2j+3\le18$

• $2j-3\le18$

• $3j+3\le18$

• $3j-3\le18$

Multiple Choice

What is the greatest age Jimmy could be?

• $j\le4$

• $j\le5$

• $j\le6$

• $j\le7$

• $j\le8$

Brenda has $𝟓𝟎𝟎 in her bank account. Every week she withdraws $𝟒𝟎 for miscellaneous expenses. How many weeks can she withdraw the money if she wants to maintain a balance of a least $𝟐𝟎𝟎?

Write and solve the inequality

Multiple Choice

Which inequality should you use?

• $500+40w\ge200$

• $500-40w\ge200$

• $500+40w\le200$

• $500-40w\le500$

Multiple Select

What is the solution and did it preserve or reverse the inequality symbol?

• $w\le7.5$

• $w\ge7.5$

• preserved

• reversed

The End