Inequalities and Their Graphs

Inequalities and Their GraphsLesson 3 1 Problem 1What inequality represent the verbal expression?All real numbers x less than or equal to -7

x -7

6 less than a number k is greater than 13

k 6 > 13Solution of InequalityAny number that makes the inequality true.

Example:

x < 5, the solution are all real numbers less than 5Problem 2Is the number a solution of 2x + 1 > -3a. -3?b. -1?

2x + 1 > -32(-3) + 1 > -3-6 + 1 > -3-5 > -3Not true!2x + 1 > -32(-1) + 1 > -3-2 + 1 > -3-1 > -3TRUE!-1 is a solution to 2x + 1 > -3GraphsTurn to page 165 to see four graphs.

What does it mean to have a solid dot?

What does it mean to have an empty dot?Problem 3, 4 and 5Turn to page 166 167.

Lesson Check

Homework: 8 39 multiples of 3 Solving Inequalities Using Addition or SubtractionLesson 3 2 Addition Property of Inequality Words:Let a, b, and c be real numbersIf a > b, then a + c > b + c.If a < b, then a + c < b + c.

Example:5 > 4, so 5 + 3 > 4 + 3-2 < 0, so -2 + 1 < 0 + 1

This property is also true for and . Problem 1What are the solutions of x 15 > -12?

x 15 > -12x 15 + 15 > -12 + 15x + 0 > 3x > 3

How would you check your answer?Problem 2What are the solutions of 10 x 3

10 x 310 + 3 x 3 + 313 x

How would you graph the solutions?Subtraction Property of Inequality Words:Let a, b, and c be real numbersIf a > b, then a - c > b - c.If a < b, then a - c < b - c.

Example:-3 < 5, so -3 - 2 < 5 - 23 > -4, so 3 - 1 > -4 - 1

This property is also true for and . Problem 3What are the solutions of t + 6 > -4?

t + 6 > -4t + 6 6 > -4 6 t + 0 > -10t > -10

How would you check this answer?Problem 4The hard drive on your computer has a capacity of 120 gigabytes (GB). You have used 85 GB. You want to save some home videos to your hard drive. What are the possible sizes of the home video collection you can save?85 + v 12085 + v 85 120 85v 35The home video can be any size less than or equal to 35 GB. Homework Lesson Check

Homework: 12 40 multiplies of 4Solving Inequalities Using Multiplication or DivisionLesson 3 3 Multiplication Property of Inequality Words:Let a, b, and c be real numbers with c > 0If a > b, then ac > bc.If a < b, then ac < bc.

Let a, b, and c be real numbers with c < 0If a > b, then ac < bc.If a < b, then ac > bc.

This property is also true for and . Heres why it works: 3 > 1-2(3) ? -2(1)-6 ? -2-6 < -2

When you multiply both sides by a NEGATIVE number, the sign is switched. Problem 1What are the solutions of (x/3) < -2?

(x/3) < -23(x/3) < -2(3)x < -6

Problem 2What are the solutions of -w 3?

-w 3(-4/3)( -w) 3(-4/3)1w -4w -4

Check your answer.Division Property of Inequality Words:Let a, b, and c be real numbers with c > 0If a > b, then a/c > b/c.If a < b, then a/c < b/c.

Let a, b, and c be real numbers with c < 0If a > b, then a/c < b/c.If a < b, then a/c > b/c.

This property is also true for and . Heres why it works: 6 > 46/(-2) ? 4/(-2)-3 ? -2-3 < -2

When you divide both sides by a NEGATIVE number, the sign is switched. Problem 3You walk dogs in your neighborhood after school. You earn $4.50 per dog. How many dogs do you need to walk to earn at least $75?(cost of dogs) times (# of dogs) is at least ($)4.50d 754.50d/4.50 75/4.50d 16.666

You must walk at least 17 dogs to earn at least $75. Problem 4What are the solutions of -9y 63?

-9y 63-9y/-9 63/-9y -7Homework Lesson Check 1 6

Homework: 8 32 multiples of 4, 46, 48Concept Byte Modeling Multi-Step InequalitiesComplete 1 8 on page 185. Solving Multi-Step InequalitiesLesson 3 4 Problem 1What are the solution of 9 + 4t > 21?

9 + 4t > 219 + 4t 9 > 21 9 4t > 124t/4 > 12/4t > 3

Check you answer. Problem 2In a community garden, you want to fence in a vegetable garden that is adjacent to your friends garden. You have at most 42 ft of fence. What are the possible lengths of your garden?Perimeter = 2l + 2w Turn to page 187 to see the picture. Problem 22L + 2(12) 422L + 24 422L + 24 24 42 24 2L 182L/2 18/2L 9

The length of the garden must be 9 feet or less.

Problem 3Which is a solution of 3(t + 1) 4t -5?a. 8b. 9c. 10d. 11

3(t + 1) 4t -53t + 3 4t -5 -t + 3 -5-t + 3 3 -5 3-t -8-t(-1) -8(-1)t 8

a is the correct answer. Problem 4What are the solutions of 6n 1 > 3n + 8?

6n 1 > 3n + 86n 3n 1 > 3n + 8 3n 3n 1 > 83n 1 + 1 > 8 + 13n > 9n > 3Problem 5aWhat are the solutions of 10 8a 2(5 4a)? 10 8a 2(5 4a)10 8a 10 8a 10 8a + 8a 10 8a + 8a10 10

Since 10 10 is always true, the solutions are all real numbers.

Problem 5bWhat are the solutions of 6m 5> 7m + 7 m?

6m 5 > 7m + 7 m6m 5 > 6m + 7-5 > 7

Since this inequality is never true, then there is no solution. Homework Lesson Check: 1 8

Homework 10 50 multiples of 5, skip 45Chapter 3 Mid-Chapter QuizWorking with SetsLesson 3 5 Sets of Numbersa group of numbers

Roster form: a list of numbers

Set-builder notation: describes a set of numbers Problem 1, 2, 3 and 4Read as a group

Lesson Check: 1 6

Homework: 10 28 evensCompound InequalitiesLesson 3 6 Compound InequaltiyConsist of two inequalities joined by the words and or the word or.

Look at page 200 to see an example.Problem 1aWhat compound inequality represents the phrase?

all real numbers that are greater than -2 and less that 6n > -2 and n < 6-2< n and n< 6-2 < n < 6

Problem 1bWhat compound inequality represents the phrase?

all real numbers that are less than 0 or greater than or equal to 5

t < 0 or t 5

How can you graph this?

Problem 2What are the solutions of -3 m 4 < -1?

-3 m 4 < -1-3 m 4-3 + 4 m 4 + 41 mm 4 < -1m 4 + 4 < -1 + 4m < 3Problem 3Turn to page 202.Problem 4What are the solutions of 3t + 2 < -7 or -4t + 5 < 13t + 2 < -7 or -4t + 5 < 13t + 2 < -73t < -9t < -3-4t + 5 < 1-4t < -4t > 1t < -3 or t > 1Interval NotationParentheses: Use ( ) for a < or > symbol

Brackets: Use [ ] for a or symbol

Infinity: Use and - to indicate that the value goes on forever in a positive or negative directionProblem 5aWhat is the graph of [-4, 6)? Write this as an inequality.

-4 x < 6Problem 5aWhat is the graph of x -1 or x > 2. Write this in interval notation.

(-, -1] or (2, )Homework Lesson Check: 1 - 6

Homework: 10 26 evensAbsolute Value Equations and InequalitiesLesson 3 7 Problem 1What are the solutions of x + 2 = 9x + 2 = 9x + 2 2 = 9 2 x = 7

x can be 7 or -7Problem 2Starting from 100 ft away, your friend skates toward you and then passes by you. She skates at a constant speed of 20 ft/s. Her distance d from you in feet after t seconds is given by d = 100 20t. At what times is she 40 ft from you?

100 20t = 40-20t = -60t = 3100 20t = -40-20t = -140t = 7Problem 3What are the solutions of 22z + 9 + 16 = 10?

22z + 9 + 16 = 1022z + 9 + 16 - 16 = 10 16 22z + 9 = -62z + 9 = -3

The absolute value can not be negative, so there is no solution.Solving Absolute Value InequalitiesIf A < b and b is positive, then solve the compound inequality

-b < A < b

If A > b and b is positive, then solve the compound inequality

A < -b or A > bProblem 4What are the solutions of 8n 24?

8n -24or 8n 248n -24n -38n -24n 3Problem 5What the solutions to the inequality?

w - 213 5-5 w 213 5-5 + 213 w 213 + 213 5 + 213208 w 218

The solution is any real number less than or equal to 213 and greater than or equal to 208. Homework Lesson Check 1 7

Unions and Intersects or SetsLesson 3 8 Turn to page 214Homework Lesson Check 1 5

Homework: 10 35 multiples of 5, 24