1. Chapter 1 Expressions, Equations and Inequalities

2. Getting Ready! You can use emojis in texts to help communicate things. How can you describe the set that includes 5 of the emojis but not the sixth? Numbers can also be classified into sets where they all have similar qualities, just like the emojis.

3. 1.2 Properties of Real Numbers Objectives: 1. To graph and order real numbers 2. To identify properties of real numbers.

4. 1. ___________________ is the set of numbers that can be expressed as a fraction. Can also be seen as an integer, a decimal that repeats or terminates. 2. __________ is the set of positive and negative whole numbers. 3. ___________ is the set of numbers used to count naturally. Also known as “counting numbers.” 4. _________________ is the set of numbers that are compromised of all rational and irrational numbers. Every number that you know of at this point is this type.

5. 5. ___________________ is the opposite of any number. 6. _______________________ is a number that cannot be expressed as a fraction. These numbers are decimals that do not repeat or terminate. 7. _________________________ is the reciprocal of any number. 8. __________________ is the set of natural numbers and zero. 9. An ____________ is comparison between two sets of numbers that are found to be not equal.

6. Without looking… label each set of Real Numbers with its proper classification name. Choose from: Rational, Whole, Integers, Irrational, Natural

7. Questions for understanding 1. What would be the sum of a number and it’s additive inverse? (i.e. 4 + (-4) = ?) 2. What would the product of a number and its multiplicative inverse? 3. What would be the additive inverse of zero? The multiplicative inverse of zero? 4. What are some examples of inequality signs? How do you graph on a number line? The sum would always be zero. They cancel each other out. The product would be 1. I.E. 1/5 times 5. Additive inverse of zero is zero! Zero does not have a multiplicative inverse! >, <, ≥, ≤, ≠

8. Questions for Understanding 1. Multiple Choice: Your school is sponsoring a charity race. Which set of numbers does contain the number of people p who participate in the race? a. Natural numbers b. Integers c. Rational numbers d. Irrational numbers 2. If each participant made a donation d of $15.50 to a local charity, which subset of real numbers best describes the amount of money raised?

9. Problem 2: Graphing Numbers on a Number Line

10. Problem 3: Ordering Real Numbers

11. Order of Operations Review O P E M D A S O 1 st – Parentheses () [] {} O 2 nd – Exponents O 3 rd – Multiply or Divide from left to right! O 4 th – Add or Subtract from left to right!

13. 1.4 Solving Equations O An _______________________ is a statement that two expressions are equal. O You can use the properties of _____________________.

14. Properties of Equality

15. Problem 1 – Solving a one- step equation

16. Problem 2: Solving a multi- step equation

17. Problem 3: Using an equation to solve a problem

18. Objectives: 1. To solve and graph inequalities. 2. To write and solve compound inequalities. 1.5 Solving Inequalities

19. O Phrases like “greater than” and “less than” suggest that two quantities may not be equal. You can represent such a relationship with a mathematical inequality statement. O Essential Understanding – You can apply the strategies used in solving equations to solving inequalities.

20. InequalityWord SentenceGraph Writing and Graphing Inequalities More than 4 At least 4 4 or more Less than 4 At most 4 No more than 4

21. O In the graphs on previous slide, the point at 4 is a BOUNDARY POINT because it separates the graph of the inequality from the rest of the number line. O An open dot at 4 means that 4 is not a solution of the inequality. O A closed dot at 4 means that 4 is a solution. Problem 1 – writing an Inequality from a sentence What inequality represents the sentence, “5 fewer than an number is at least 12.”? Got It? What inequality represents the sentence, “The quotient of a number and 3 is no more than 15.”?

22. O The properties you use for solving inequalities are similar to properties you use for solving equations. HOWEVER, when you multiply (or divide) an inequality by a negative number, YOU MUST REVERSE the inequality symbol.

23. Here’s Why It Works The steps below show that if a 3. Now multiply both sides by negative one, what happens? Does the “>” sign keep a true statement?

24. What is the solution of -3 (2x – 5) + 1 ≥ 4 Got It?What is the solution of each inequality? (a) 2x + 10 ≤ 2 (b)2(4x + 1) + 3 > -3 Problem 2 – Solving and Graphing an Inequality

25. Problem 3 – Using an Inequality Movie Rentals A movie rental company offers two subscription plans. You can pay $36 a month and rent as many movies as desired, or you can pay $15 a month and $1.50 to rent each movie. How many movies must you rent in a month for the first plan to cost less than the second plan? Let’s use x to represent the number of movies you might rent each month. You can write this problem as an inequality. The left side can represent the cost of the 1 st plan and the right side can represent the cost of the 2 nd plan. 36 < 15 + 1.50x (now solve for x) (graph your solution on a number line)

26. Got it?A digital music service offers two subscription plans. The first has a $9 membership fee and chares $1 per download. The second has a $25 membership fee and charges $0.50 per download. How many songs must you download for the second plan to cost less than the first plan?

27. Problem 4 – No Solution or All Real Numbers as a Solution Is the inequality always, sometimes, or never true? (a) -2(3x + 1) > -6x + 7(b)5(2x – 3) – 7x ≤ 3x + 8 (c) 6(2x – 1) ≥ 3x + 12

28. Got it? Is the inequality always, sometimes, or never true? (a) 4(2x – 3) < 8(x + 1)(b) –6(4x – 3) – 3 ≥ -9 (c) 6(2x – 1) – 8x > 3 + 4x

29. Essential Understanding – You can join two inequalities with the word and or the word or to form a compound inequality. To solve a compound inequality containing AND you find ALL the values of the variable that make both inequalities true. Problem 5 – Solving an AND Inequality. What is the solution of 2x + 1 > 7AND 3x ≤ 18? Graph the solution. Got It? What is the solution of 3x – 1 ≥ 5 AND 2x < 12? Graph the solution.

30. Sometimes, you can combine a compound and inequality like and, into a simpler form:

31. To solve a compound inequality containing or, find ALL values of the variable that make at least one of the inequalities true. Problem 6 – Solving an OR Inequality What is the solution of 7 + x ≥ 6 OR 8 + x < 3 ? Graph the solution. Got It? What is the solution of 7x + 3 > 11 OR 4x – 1 < -13? Graph the solution.

33. Objectives: To write and solve equations and inequalities involving absolute value. 1.6 Absolute Value Equations and Inequalities

34. Essential Understanding- O An absolute value quantity is NON-NEGATIVE. Since opposites have the same absolute value, an absolute value equation can have TWO solutions. Key Concept: Absolute Value O The ABSOLUTE VALUE of a real numbers x, written | x |, is its distance from ZERO on the number line. Examples: NUMBERSSYMBOLS |4| =|x| =, if x > |-4| =|-x| =, if x <

35. You MUST satisfy 2 conditions BEFORE solving: O 1. Make sure the absolute value expression is completely alone on one side of the equation!! O Nothing “in front, behind or underneath” O 2. Make sure the number on the other side is POSITIVE!! O If the number is NEGATIVE then there is NO SOLUTION!!!! You must be able to say YES to BOTH conditions!!!!!

36. To solve…break your “new” equation into 2 different ones: O 1. Remove the bars and copy the problem exactly as written!! O 2. Remove the bars and set the equation equal to its opposite sign!!

37. Problem 1: Solving an Absolute Value Equation a.What is the solution of |2x – 1| = 5? Solve for x and graph the solution. b.What is the solution of |3x + 2| = 4? Solve for x and graph the solution.

38. Problem 2: Solving a Multi-Step Absolute Value Equation a.What is the solution of 3|x + 2| - 1 = 8 Solve for x and graph the solution. b.What is the solution of 2|x + 9| + 3 = 7? Solve for x and graph the solution.

39. Key Concept : Extraneous Solutions O Distance from zero on the number line cannot be NEGATIVE. Therefore, some absolute value equations, such as | x | = -5, have NO SOLUTION. It is important to check the possible solutions of an absolute value equation. O One or more of the possible solutions may be EXTRANEOUS. Definition - O An EXTRANEOUS SOLUTION is a solution derived from an original equation that is not valid.

40. Problem 3: Checking for Extraneous Solutions a.What is the solution of |3x + 2| = 4x + 5? Check for extraneous solutions. b.What is the solution of |5x - 2| = 7x + 14? Check for extraneous solutions.

41. Key Concept : Compound Inequality “and” with Abs. Value Essential Understanding - O You can rewrite an absolute value inequality as a compound inequality WITHOUT absolute value symbols. O Problem 4: Solving the Absolute Value Inequality | A | < b a.What is the solution of |2x - 1| < 5? Graph the solution. b.What is the solution of |3x - 4| < 8? Graph the solution.

42. Key Concept : Compound Inequality “OR” with Abs. Value Problem 5: Solving the Absolute Value Inequality | A | > b a.What is the solution of |2x + 4| ≥ 6? Graph the solution. b.What is the solution of |5x + 10| > 15? Graph the solution.