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Thursday, september 26, 2013

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Thursday, september 26, 2013

1. 1. September 26, 2013 • Today: • Make Up Tests? • Grades shown Today, questions tomorrow • Warm- Up: Equations • The real number system: Integers & Absolute Value
2. 2. • Warm-Up, Equations: • 1. A. 2 + 5 = 7; Solve for 5; 5 = B. Solve for 2; 2 = • C. 14 - 8 = 6; Solve for -8; -8 = D. Solve for 14; 14 = • E. 12/4 = 3; Solve for 4 F. Solve for 12; 12 = • G. (5)(3) = 15; Solve for 5; 5 = H. Solve for 3; 3 = The rules are the same for variables, because variables are just numbers we haven't figured out yet. 2. Write 4 different equations so that the answer to each one is x = 5. The first equation should have a sum, the second a difference, the third a product and the fourth a quotient. x + 5 = 7; x = 7 – 5; x = 2
3. 3. Simplify the following: 1. -12c + 3 - 9(11 - c) 2. -.3x - 4.2 + 6.1x - .9 Warm-Up, Con’t:
4. 4. Vocabulary & Formulas
5. 5. Natural Numbers: {1, 2, 3, …} Whole Numbers: {0, 1, 2, 3, …} Integers: {…, -2, -1, 0, 1, 2, …} Rational Numbers: Numbers expressed in the form a/b, where a and b are integers and b ≠ 0 The Real Numbers Includes all fractions, Mixed numbers, Ratios, Proportions, & decimals. Irrational Numbers: These numbers don't end and they don't repeat. Ex. √2, Pi,
6. 6. Real Numbers Practice: For each of the numbers, write Natural, Whole, Integer, Rational, or Real according to its type. Most will have more than one classification. 1. 0.25 2. 8.25252525 3. -1/2 4. 5 5. 0 6. -5 7. √5 8. 200 ft. below sea level
7. 7. Integer Rules: 1. For Multiplying : If there is an odd number of negative values, the product will be negative. If the number of values is even, the product will be negative: 3(-5) 3(-5)(-2) 3(-5)(-2)(-3) 2. For Dividing : Since we can only divide two numbers at a time, the rule is easy: If one of the numbers is negative, the quotient is negative. If both are positive, or both are negative, the quotient is positive. -28 ÷ (-4) 35 ÷ -7 -25 ÷ -5 -4ab•(-6) (-4)(-2)(8r)
8. 8. Integer Rules: 3. For Adding & Subtracting 3a. If the signs are the same, add the values and use the given sign. 3b. If the signs are different, subtract the values and use the sign of the larger number. 3c. Sometimes you must reduce the number of signs before you add or subtract. 1. -2 - (-19) ALWAYS leave the sign of the first number alone. The first number shows the starting point of the calculation Our focus is always on the next two signs. If they are the same, change them both to a single plus + sign. If they are different, change them to a single negative sign. 1. 2 + 5 2. -2 - 5
9. 9. Integer Rules: At this point, the original expression -2 - (-19) Now looks like: -2 +19 Only now can we go back to our original rules, 3a. And 3b. (Signs are the same or signs are different) 1. -7 + - 5 2. - 8 + - 10 + 8 3. - 4 + (- 5) + 6 4. - 2 - (- 8) 5. - 4 - (- 10) 6. 4 - 22 1. - 12 2. - 10 3. - 3 4. 6 5. 6 6. - 18
10. 10. Integer Rules: --14 12 15 Challenge:
11. 11. Absolute Value: |x| • An important concept in Algebra; one that you should try to master. Absolute Value measures the distance a number is from zero. The following are illustrations of what absolute value means using the numbers 3 and -3: Since Absolute Value is a measure of distance, the result can never be negative. (There can be no negative distances)
12. 12. • If a number is positive (or zero), the absolute value function does nothing to it: |4| = 4 • If a number is negative, the absolute value function makes it positive: |-4| = 4 Find the value of the following: |5 + (-2)| Did you get 7? Unfortunately, that's wrong. If there is arithmetic to do inside the absolute value sign, you must do it before taking the absolute value sign. The correct answer is: |5 + (-2)| = |3| = 3 The following has no solution: | x |= -5 Why? Absolute Value: |x|
13. 13. • Simplify | 0 – 6 | • Simplify | 4+ (– 6)| • Simplify | 2 – 5 | • Simplify | 0(–4) | • Simplify | 2 + 3(–4) | • Simplify –| –4 | • Simplify –| (–2)2 | • Simplify –| –2 |2 Absolute Value: |x|
14. 14. Class Work: See Instructions on Handout
15. 15. 6 + 9 + 2 = 17; -3 + (-5) + (-4) = -12; 17 - 12 = 5 Use Method B: 5 + (-4) + (-6) + 8 + (-3) + 3 = 1 -6 = -5 + 8 = 13 - 3 = -0 + 3 = 13 1. There are 2 ways to combine long strings of integers. A. Add all the positive numbers, then add all the negative numbers. Then combine the two numbers using rule for opposite signs. B. Combine all numbers in order from left to right. Use method A for these numbers: -3 + (-5) + 6 + 9 + 2 + (-4)