Lesson 23, Section 4 - Purdue University



Lessons 21, Section 4.1

Inequalities

Ex 1: Which of the following numbers would be solutions of the inequality shown?

[pic] 0, -5, -3, 12

[pic] true

[pic] false

[pic] true

[pic] true

Solutions include 0, -3, and 12.

There are three ways to represent an inequality; using the inequality symbol (set-builder notation), a number-line graph, and using interval notation. The examples below represent equivalent forms of all three ways.

inequality symbol number-line graph interval notation

[pic] [pic]

5

[pic] [pic]

2

[pic] (0, 3)

0 3

[pic] [4, 10]

4 10

[pic] [pic]

-3 1

There is a summary of these ways on the course webpage (other information, inequalities).

*Do not confuse interval notation with an ordered pair (point). The context in which each is used will make the meanings clear.

a) Write in interval notation and graph on the number line. [pic]

b) Write using set-builder notation and using interval notation.

-5

c) Write using set-builder notation and graph on the number line. [pic]

Begin with the following inequality: 20 > 12 Do the following operations to both sides of the inequality and determine if the result is true or false.

add 4 20 + 4 > 12 + 4 true

subtract 3 20 - 3 > 12 - 3 true

multiply by 4 4(20) > 4(12) true

divide by 2 [pic] true

multiply by -2 -2(20) > -2(12) ?

divide by -4 [pic] ?

Solving Inequalities: When solving an inequality you may add, subtract, multiply by a positive number, or divide by a positive number on both sides and the result is true. However, if you multiply or divide by a negative number, the inequality sign must be reversed!

Solve these inequalities. Write the answer using both set-builder notation and interval notation, then graph the solution.

Ex 2: [pic]

Ex 3: [pic]

Ex 4: [pic]

Ex 5: [pic]

Ex 6: [pic]

Ex 7: [pic]

Ex 8: [pic]

Ex 9: [pic]

Applications of Inequalities

Phrases commonly translated to inequalities:

|Words |Sample Sentence |Translation |

|is at least |Gina is at least 18 years old |[pic] |

|are at most |There are at most 12 coins |[pic] |

| |in his pocket. | |

|cannot exceed |The total value cannot exceed $12. |[pic] |

|must exceed |The car's speed must exceed 20 miles per |[pic] |

| |hour | |

|no more than |There can be no more than 50 people in the |[pic] |

| |room. | |

|no less than |The nut mix can have no less than 5 cups of|[pic] |

| |almonds. | |

|is less than |Marie's age is less than 21 |[pic] |

|is more than |The house is more than 35500 square feet. |[pic] |

|is between |The distance is between 50 and 200 miles. |[pic] |

Examples:

1) Metro Concerts can rent a truck for either $55 with unlimited mileage or $29 plus

40¢ per mile. For what mileages would the unlimited mileage plan save money?

Let m = numbers of miles

2) A long-distance phone call using East Calling costs 10 cents for the first minute and 8 cents for each additional minute. The same call on West Call System cost 15 cents for the first minute and 6 cents for each additional minute. For what length phone calls is East Calling less expensive than West Call System?

Let m = number of total minutes

3) Musclebound Movers charges $85 plus $40 an hour to move households across town. Champion Movers charges $60 an hour for cross-town moves. For what lengths of time is Champion more expensive?

Let h = number of hours for the move

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Note: In place of an open circle, we now use a parenthesis. In place of a closed circle, we now use a brachet. Both are to open in the direction of the shading.

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