Lesson 3 – 1

What inequality represent the verbal expression? a. All real numbers x less than or equal to -7 x ≤ -7 b. 6 less than a number k is greater than 13 k – 6 > 13

Any number that makes the inequality true. Example: x < 5, the solution are all real numbers less than 5

Is the number a solution of 2x + 1 > -3 a. -3?b. -1? 2x + 1 > -3 2(-3) + 1 > > > -3 Not true! 2x + 1 > -3 2(-1) + 1 > > > -3 TRUE! -1 is a solution to 2x + 1 > -3

 Turn 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?

 Turn to page 166 – 167.  Lesson Check  Homework: 8 – 39 multiples of 3

Lesson 3 – 2

Words: Let a, b, and c be real numbers If a > b, then a + c > b + c. If a < b, then a + c < b + c. Example: 5 > 4, so > < 0, so < This property is also true for ≤ and ≥.

 What are the solutions of x – 15 > -12? x – 15 > -12 x – > x + 0 > 3 x > 3 How would you check your answer?

 What are the solutions of 10 ≥ x – 3 10 ≥ x – ≥ x – ≥ x How would you graph the solutions?

Words: Let a, b, and c be real numbers If a > b, then a - c > b - c. If a < b, then a - c < b - c. Example: -3 < 5, so < > -4, so > This property is also true for ≤ and ≥.

 What are the solutions of t + 6 > -4? t + 6 > -4 t + 6 – 6 > -4 – 6 t + 0 > -10 t > -10 How would you check this answer?

 The 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 ≤ v – 85 ≤ 120 – 85 v ≤ 35 The home video can be any size less than or equal to 35 GB.

 Lesson Check  Homework: 12 – 40 multiplies of 4

Lesson 3 – 3

Words: Let a, b, and c be real numbers with c > 0 If a > b, then ac > bc. If a < b, then ac < bc. Let a, b, and c be real numbers with c < 0 If a > b, then ac < bc. If a bc. This property is also true for ≤ and ≥.

3 > 1 -2(3) ? -2(1) -6 ? < -2 When you multiply both sides by a NEGATIVE number, the sign is switched.

What are the solutions of (x/3) < -2? (x/3) < -2 3(x/3) < -2(3) x < -6

What are the solutions of -¾w ≥ 3? -¾w ≥ 3 (-4/3)( -¾w) ≥ 3(-4/3) 1w ≤ -4 w ≤ -4 Check your answer.

Words: Let a, b, and c be real numbers with c > 0 If 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 < 0 If a > b, then a/c < b/c. If a b/c. This property is also true for ≤ and ≥.

6 > 4 6/(-2) ? 4/(-2) -3 ? < -2 When you divide both sides by a NEGATIVE number, the sign is switched.

 You 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 ≥ d/4.50 ≥ 75/4.50 d ≤ You must walk at least 17 dogs to earn at least $75.

 What are the solutions of -9y ≤ 63? -9y ≤ 63 -9y/-9 ≥ 63/-9 y ≥ -7

 Lesson Check 1 – 6  Homework: 8 – 32 multiples of 4, 46, 48

Complete 1 – 8 on page 185.

Lesson 3 – 4

What are the solution of 9 + 4t > 21? 9 + 4t > t – 9 > 21 – 9 4t > 12 4t/4 > 12/4 t > 3 Check you answer.

In a community garden, you want to fence in a vegetable garden that is adjacent to your friend’s 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.

2L + 2(12) ≤ 42 2L + 24 ≤ 42 2L + 24 – 24 ≤ 42 – 24 2L ≤ 18 2L/2 ≤ 18/2 L ≤ 9 The length of the garden must be 9 feet or less.

Which is a solution of 3(t + 1) – 4t ≥ -5? a. 8b. 9c. 10d. 11 3(t + 1) – 4t ≥ -5 3t + 3 – 4t ≥ -5 -t + 3 ≥ -5 -t + 3 – 3 ≥ -5 – 3 -t ≥ -8 -t(-1) ≤ -8(-1) t ≤ 8 a is the correct answer.

What are the solutions of 6n – 1 > 3n + 8? 6n – 1 > 3n + 8 6n – 3n – 1 > 3n + 8 – 3n 3n – 1 > 8 3n – > n > 9 n > 3

What are the solutions of 10 – 8a ≥ 2(5 – 4a)? 10 – 8a ≥ 2(5 – 4a) 10 – 8a ≥ 10 – 8a 10 – 8a + 8a ≥ 10 – 8a + 8a 10 ≥ 10 Since 10 ≥ 10 is always true, the solutions are all real numbers.

What are the solutions of 6m – 5> 7m + 7 – m? 6m – 5 > 7m + 7 – m 6m – 5 > 6m > 7 Since this inequality is never true, then there is no solution.

 Lesson Check: 1 – 8  Homework 10 – 50 multiples of 5, skip 45

Lesson 3 – 5

a group of numbers Roster form: a list of numbers Set-builder notation: describes a set of numbers

Read as a group Lesson Check: 1 – 6 Homework: 10 – 28 evens

Lesson 3 – 6

 Consist of two inequalities joined by the words and or the word or.  Look at page 200 to see an example.

 What compound inequality represents the phrase? a. all real numbers that are greater than -2 and less that 6 n > -2 and n < 6 -2< n and n< 6 -2 < n < 6

 What compound inequality represents the phrase? b. 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?

What are the solutions of -3 ≤ m – 4 < -1? -3 ≤ m – 4 < ≤ m – ≤ m – ≤ m m – 4 < -1 m – < m < 3

Turn to page 202.

What are the solutions of 3t + 2 < -7 or -4t + 5 < 1 3t + 2 < -7 or -4t + 5 < 1 3t + 2 < -7 3t < -9 t < -3 -4t + 5 < 1 -4t < -4 t > 1 t 1

 Parentheses: Use ( ) for a symbol  Brackets: Use [ ] for a ≤ or ≥ symbol  Infinity: Use ∞ and -∞ to indicate that the value goes on forever in a positive or negative direction

What is the graph of [-4, 6)? Write this as an inequality. -4 ≤ x < 6

What is the graph of x ≤ -1 or x > 2. Write this in interval notation. (-∞, -1] or (2, ∞)

 Lesson Check:  Homework: 10 – 26 evens

Lesson 3 – 7

 What are the solutions of │ x │ + 2 = 9 │ x │ + 2 = 9 │ x │ + 2 – 2 = 9 – 2 │ x │ = 7 x can be 7 or -7

Starting 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 = t = -60 t = – 20t = t = -140 t = 7

What are the solutions of 2 │ 2z + 9 │ + 16 = 10? 2 │ 2z + 9 │ + 16 = 10 2 │ 2z + 9 │ = 10 – 16 2 │ 2z + 9 │ = -6 │ 2z + 9 │ = -3 The absolute value can not be negative, so there is no solution.

If │ 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

What are the solutions of │ 8n │ ≥ 24? 8n ≤ -24or 8n ≥ 24 8n ≤ -24 n ≤ -3 8n ≥ -24 n ≥ 3

What the solutions to the inequality? │ w │ ≤ 5 -5 ≤ w – 213 ≤ ≤ w – ≤ ≤ w ≤ 218 The solution is any real number less than or equal to 213 and greater than or equal to 208.

 Lesson Check 1 – 7

Lesson 3 – 8

 Lesson Check 1 – 5  Homework: 10 – 35 multiples of 5, 24