Introduction to Sets, Operations, and Equations in Mathematics

1. Chapter 1 Set: a collection of objects called elements Important sets of numbers: Natural numbers Whole numbers Integers

2. Use the ‘roster method’ to write a set Inequality symbols Additive inverses Absolute value

3. Addition of integers • SAME SIGNS 1. Add the absolute values of the numbers ( ignore the signs and add) 2. Attach the common sign

4. DIFFERENT SIGNS 1. Find the difference of the absolute values (ignore the signs and subtract) 2. Attach the sign of the number with the larger absolute value.

5. Subtraction of integers 1. Rewrite the “—” as “+ the opposite of the number” 2. Follow the rules for addition.

6. Subtraction of integers • We might say this subtraction process as “Change to the opposite then add”

7. Multiplication of integers • The product of two numbers with the same sign is positive. positive · positive = + negative · negative = +

8. Multiplication of integers • The product of two numbers with different signs is negative. positive · negative = negative negative · positive = negative

9. When multiplying or dividing: ODD # of negative signs makes the answer negative. EVEN # of negative signs makes the answer positive.

10. Division of integers (The rules are like those for multiplication) • The quotient of two numbers with the same sign is positive. positive ÷ positive = + negative ÷ negative = +

11. The quotient of two numbers with different signs is negative positive ÷ negative = negative negative ÷ positive = negative

12. If a fraction is negative, the ‘–’ sign can be place in any of three different positions, and all are considered equivalent.

13. 0 ÷ any number = 0 • Division by 0 is not defined! • To compute the arithmetic mean or average:

14. Operations with rational numbers • Rational numbers are fractions and decimals which repeat or stop. • Reduce fractions to simplest form • Change fractions into decimals which repeat or stop

15. Add, subtract, multiply and divide fractions • Add, subtract, multiply and divide decimals • Don’t forget to use our rules for + and - signs!

16. Working with % • Change % to a decimal • Change a decimal to a % • Change a fraction to a % • Change a % to a fraction

17. Order of Operations 1. Do operations within grouping symbols first. 2. Exponents 3. Multiply and divide from left side of problem toward right side. 4. Add and subtract last!

18. Working with Exponents 24 = (2)(2)(2)(2) ( –2)4 is not the same as – 24

19. A good way to remember the order is PEMDAS Remember the “parens” can also be brackets, absolute value symbols, or a fraction bar.

20. Chapter 2 • Variable: a letter used to stand for a quantity • A variable expression is made up of terms

21. Types of terms: variable terms constant terms A variable term has a coefficient

22. When evaluating variable expressions, remember PEMDAS to get the right order of operations. Also, watch your signs!

23. Like terms – terms which have the exact same variable part Constant terms are also “like terms”

24. Only ‘like’ terms can be added or subtracted!!!

25. Be able to use the distributive property

26. Translating Verbal Expressions See p. 67 for a list of common phrases…..

27. When you see the phrase “in terms of”… let the part which follows be x

28. Chapter 3 Solve an equation means to find the value which makes the equation true. We want to end with : Variable = constant

29. To do this, perform ‘opposite operations’ to both sides of the equation. • On your assignments, please show the process!

30. To check whether a value is really a solution to an equation, put it in place of the variable and see if it makes a true statement.

31. % Problems • Change % to a decimal • Of  multiply • Is  = • Write an equation • Solve the equation

32. Solving Equations 1. First simplify each side of the equation. Distribute to get rid of parens and combine like terms

33. 2. Add or subtract terms to move all constants to one side and all variable terms to the other side of the =

34. 3. Divide or multiply to get rid of the coefficient of the variable. End with : variable = constant

35. 3.4 Translating Sentences into Equations These words or phrases are replaced with an = sign: equalsis is equal to amounts to represents

36. Steps: 1. Assign a variable to the unknown quantity. 2. Translate the words into math symbols. Write the equation. 3. Solve the equation. 4. Check your answer.

37. Recall the integers are the positive and negative whole numbers: {… -4, -3, -2, -1, 0, 1, 2, 3, 4 …} An even integer is an integer that is divisible by 2 like 12, -4, 28, 0 An odd integer is not divisible by 2 like 33, -27, and 5

38. Consecutive Integers(none in exercises) 5, 6, 7 or -11, -10, -9 or n, n + 1, n + 2 Consecutive Even Integers -12, -10, -8 or 4, 6, 8 or n, n + 2, n + 4 Consecutive Odd Integers 5, 7, 9 or -13, -11, -9 or n, n + 2, n + 4

39. Find three consecutive even integers such that three times the second is four more than the sum of the first and the third.

40. Five times the first of two consecutive odd integers equals three times the second integer. Find the integers.

41. Translate “three more than twice a number is the number plus six” into an equation.

42. Four less than one-third of a number equals five minus two-thirds of the number. Find the number

43. The sum of two numbers is sixteen. The difference between four times the smaller number and two is two more than twice the larger number. Find the two numbers.

44. The sum of two numbers is twelve. The total of three times the smaller number and six amounts to seven less than the product of four and the larger number. Find the two numbers.

45. The difference between a number and twelve is twenty. Find the number.

46. A board 20 ft long is cut into two pieces. Five times the length of the smaller piece is 2 ft more than twice the length of the longer piece. Find the length of each piece.

47. A company makes 140 televisions per day. Three times the number of black and white TV’s made equals 20 less than the number of color TV’s made. Find the number of color TV’s made each day.

48. Translating Sentences into Equations 1. Assign a variable or an expression to the unknown quantity or quantities. 2. Translate the verbal expressions into math symbols. We want two expressions equal to each other. 3. Solve the equation. 4. Check your answer.

49. Chapter 4 • Monomial….a number, a variable, or the product (mult.) of numbers and variables • Polynomial…2 or more monomials added or subtracted (The monomials are then called terms.)

50. Special polynomials • Binomial: 2 terms • Trinomial: 3 terms