NOTES: SIGNED INTEGERS DAY 1

Transcription

1 NOTES: SIGNED INTEGERS DAY 1 MULTIPLYING and DIVIDING: Same Signs (POSITIVE) + + = + positive x positive = positive = + negative x negative = positive Different Signs (NEGATIVE) + = positive x negative = negative + = negative x positive = negative Consecutive signs are two signs next to each other. Consecutive signs follow the same rules as multiplying and dividing. ADDING and SUBTRACTING: Same Signs (ADD) Both Positive: Add the numbers. The answer is positive. Both Negative: Add the numbers. The answer is negative. Different Signs (SUBTRACT) Subtract the numbers (find the difference). The answer is the sign of the larger number Simplify: = = = = = =

2 NOTES: EVALUATING FUNCTIONS How do we evaluate functions? If we are given a function 3x-8, and we want to know the value of the function when x = 3, we replace each x in the function with 3. Instructions Steps 1. Original problem. 3x 8, x = 3 2. Substitute 3 for x 3( ) 8 3. Simplify. Evaluate the expressions. 1. 2x + 7, given that x = x 2 3, given that x = a 2b + 4c 2 where a = 5, b = 4, c = [(13 - x) + 5y] z 2 where x = 3, y = -1, z = 2

3 NOTES: FRACTIONS DAY 2 1. Convert Mixed Number to Fraction 1. Multiply denominator by whole part. 2. Add the numerator to your first product. 3. Rewrite the fraction as improper by placing the number from step 2 over the same denominator as the original mixed number Example 1: Example 2: Multiplying Fractions 1. Change all mixed numbers to improper fractions 2. Cross-Reduce if possible 3. Multiply Top x Top 4. Multiply Bottom x Bottom 5. Simplify if Possible (Change all improper fractions to mixed numbers.) Example 1: 2 3 i 6 12 Example 2: 30 7 i 14 3

4 Dividing Fractions 1. Change all mixed numbers to improper fractions 2. Keep the first fraction, change the division sign to a multiplication sign, change the second fractions to its reciprocal 3. Cross-Reduce if possible 4. Multiply Top x Top 5. Multiply Bottom x Bottom 6. Simplify if Possible

5 NOTES: FRACTIONS DAY 2

6 Adding Fractions: 1. Do NOT change mixed numbers to improper fractions 2. Find a common denominator 3. Create equivalent fractions with the common denominator 4. Add the numerator 5. Denominator stays the same 6. Simplify Example 1: Example 2: Subtracting Fractions 1. Do NOT change mixed numbers to improper fractions 2. Find a common denominator 3. Create equivalent fractions with the common denominator 4. Subtract the numerator 5. If you need to borrow (only for the top fraction): Step 1:Subtract one from your whole number Step 2: Add your denominator to your numerator 5. Denominator stays the same 6. Simplify Example 3: Example 4:

7 NOTES: ORDER OF OPERATIONS DAY 2 Why do we even need PEMDAS? Evaluate this: Parentheses ( ) (what s INSIDE) Exponents 2 (and roots ) Multiply x Divide Addition + Subtraction Evaluate each expression WITHOUT a calculator. Reduce any fractions (3 5) (2 5) Where would you place parentheses so that the expression evaluates to 0?

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9 NOTES: COMBINE LIKE TERMS DAY 3 Definition Term: Terms are separated by addition and subtraction. Example: 2x + 3y 7 This expression has 3 terms Combining Like Terms Example 1: 3 + 5x 4 + 2x Example 2: 4x x x + 2x 4x 2 x x 3x Combine Like Terms. 1. 3xy + 5 2xy y y (5 x) + 12x x x 5. 9 (11 x) = 3(3x 9) 6. 5 (x + 2) = 22 3x 7. x ( 5)=5 8. 4x 8 = x+1 3x = 16

10 NOTES: SOLVING EQUATIONS DAY 3 Steps: 1. Eliminate parentheses (Distribute) 2. Combine like terms (on the same side) 3. Use addition or subtraction to get the variables on the same side 4. Undo addition or subtraction 5. Undo multiplication or division Solve each equation for the given variable WITHOUT using a calculator. Possible answers could be No Solution or All Real Numbers. 1. 3x 8 = x + 5x 8 = (11 x) = 3 (3x 9) 4. 9 (11 x) = 3 (3x 9) 5. 4x + 5 (7x 3) = 9 (x 5)

11 SOLVING EQUATIONS GUIDE DAY 3 Single-Step Equation Negatives 1. The variable has a negative in front of it! x = 10 x = Multiply both sides by a negative This will change the sign of each term in the equation. Single-Step Equation Addition 1. A variable plus another value. x + 9 = The opposite of addition is subtraction. Subtract that value from both sides. 3. Simplify (rewrite) both sides. Single-Step Equation Subtraction 1. A variable minus another value. x 7 = The opposite of subtraction is addition. Add that value to both sides. 3. Simplify (rewrite) both sides. Single-Step Equation Backward Subtraction 15 x = 9 3 x = A number minus a variable. 2. We are still trying to isolate the variable. 3. What is the sign of the first number? 4. Do the opposite of that sign to both sides 5. Simplify (rewrite) both sides. If the variable is negative, do not forget the sign!

12 Single-Step Equation Multiplication 1. A variable with a number in front of it (coefficient). 3n = 15-3n = Since there is not an operation between the variable and the coefficient, it is implied multiplication. 3. The opposite of multiplication is division. Divide both sides by that value. 4. Simplify (rewrite) both sides. Single-Step Equation Division 1. A variable divided by a value The opposite of division is multiplication. Multiply both sides by that value. 3. Simplify (rewrite) both sides. Single-Step Equation Fractional Coefficient 1. A variable with a fraction in front of it. Recall, the x can be in front or on the numerator: 2. There is implied multiplication between the variable and the coefficient We would generally divide by the fraction, but 4. Dividing by a fraction is the same as multiplying by the reciprocal. 5. Multiply both sides by the reciprocal (fraction flipped upside down). If the fraction is negative, multiply by the negative too. 6. Simplify (rewrite) both sides.

13 Combining Like Terms 1. Terms are separated by addition and subtraction. 6x y = They can only be combined if they are exactly the same except the coefficient! 3. Combine like terms by adding the coefficients. 4. The variables/exponents stay the same. Distributive Property 1. If there are two or more terms (added or subtracted) inside parentheses that cannot be combined. And there is a number in front that is being multiplied. -3 (y + 4) = -6 8 (t 1) = Multiply the number in front by each of the terms inside the parentheses. 3. After you distribute there are no more parentheses! Keep track of your negatives! 4. Simplify any double signs (two signs next to each other). Proportions Variable in the Numerator 1. A fraction equal to another number or fraction. The variable is on the numerator. 10 Remember: a number can always be turned into a fraction by putting 1 in the denominator. 2. Isolate the variable by multiplying both sides by the reciprocal of the coefficient. 3. On one side you may need to multiply two fractions. Reduce first! Remember: reducing can take place on the diagonals or on a fraction.

14 Proportions Variable in the Denominator A fraction equal to another number or fraction. The variable is in the denominator. 2. Take the reciprocal of both sides (flip both sides) 3. Solve the proportion as explained above. Proportions Two terms (binomials) A proportion with two unlike terms in the numerator. 2. Multiply both sides by the denominator. 3. Solve the multi-step equation that is left over. Proportions Two terms (binomials) A proportion with two unlike terms in the denominator. 2. Take the reciprocal of both sides (flip both sides). 3. Solve the proportion as explained above. Multi-Step General 1. More than one operation is being done to the variable. 2x + 3 = 7-4x + (-3) = Eliminate any two signs next to each other (consecutive) Eliminate any parentheses by combining like terms. 4. Add or subtract (the opposite) to both sides to eliminate a term. 5. Multiply or divide to both sides to eliminate the coefficient. 6. Simplify again, if necessary

15 Multi-Step Several Terms 1. There are several terms and operations being done to the variable (-3x) + 1 = Eliminate any two signs next to each other (consecutive). 3. Eliminate any parentheses by combining like terms. 4. Combine all like terms on the right side. Then combine all like terms on the left side. 5. There should be only 3 terms now and one with a variable. 6. Add or subtract (the opposite) to both sides to eliminate a term. 7. Multiply or divide to both sides to eliminate the coefficient. 8. Simplify again, if necessary Multi-Step Distributive Property 1. There are several terms and operations being done to the variable. There are also parentheses with terms that cannot be combined. 2. Distribute. After you distribute, there are no more parentheses! 3. Combining any like terms on the right side. Combine any like terms on the left. 4. There should be only 3 terms now and one with a variable. 5. Solve the equation using previous instructions. 3x x = (x 3) = 4 3y 2 (3 y) = 12

16 Multi-Step Variables on Both Sides 1. There are variables on both sides of the equation! 2c + 6 = 5c 3 6y + 3 = 8y Simplify the right side by eliminating parentheses and combining like terms. Simplify the left side also Move one of the variable terms (smaller one is sometimes easier) from one side to the other by adding or subtracting it from both sides. 4. Now there is only one variable term on one side of the equation. 5. Solve the equation by following previous instructions. Multi-Step Variables on Both Sides and Distributive Property! -3 (x + 2) + 3 = 5 (x 1) There are variables on both sides of the equation and you will need to distribute! 2. Distribute where necessary. 3. Simplify the right side by eliminating any parentheses and combining like terms. Simplify the left side also Move one variable term to the other side by adding or subtracting it to both sides. 5. Solve the equation by following previous instructions.

17 NOTES: ABSOLUTE VALUE DAY 4 1. Absolute value means. 2. Treat the absolute value symbols like parentheses (during order of operations). Evaluate the expressions = 4. 5 = 5. 6

18 NOTES: SQUARE ROOTS AND CUBE ROOTS DAY 4 Perfect squares: 1 1 = 1 2 = 1 1 is the square root of 1. 1 = = 2 2 = is the square root of 4 = = 4 4 = 5 5 = 6 6 = 7 7 = 8 8 = 9 9 = = = = Ex: a) 100 = b) 64 = c) 1 = d) 0 = e) 81 = Perfect Cubes: = 1 3 = 1 1 is the cube root of = = 2 3 = 8 2 is the cube root of = = = = = = = = = 3 3 Ex: a) 64 = b) 27 = c) = d) 0 = e) 8 =

19 NOTES: SIMPLIFYING SQUARE ROOTS DAY 4 Simplifying radicals using the Product Property: Why do we care?? This property helps us simplify radical expressions... = = = Steps for simplifying radicals: 1. Make a list of perfect squares and keep it in sight!!! Factor using the largest perfect square factor: = 3. Simplify: Be sure to use the largest perfect square!!!! = = You try:

20 NOTES: TRANSLATING EXPRESSIONS DAY 5 Addition Sum: the sum of 6 and a number Subtraction Difference: The difference of 3 and a number Increased by: A number increased by 1 Less Than: 4 less than a number Plus: 8 plus a number Minus: 7 minus a number More Than: 8 more than a number Decreased by: A number decreased by 9 Multiplication Product: The product of 4 and a number Division Quotient: The quotient of a number and 10 Times: 6 times a number Divided: 7 Divided by a number Multiplied by: A number multiplied by 2 Now you try! 1. The sum of three times a number b and 33: 2. The difference of 32 and the quotient of a number m and 7: Partner Activity! Create an expression for your partner to translate, and then switch. Expression: Translation:

21 NOTES: APPLICATION PROBLEMS DAY 5 1. An item costs c dollars and 6% sales tax is charged. The total cost including sales tax is given by the expression 1.06c. You are buying a skateboard that costs $75. What is the cost of the skateboard including sales tax? 2. You are creating a flower arrangement for a friend. The total cost (in dollars) for one vase and f flowers is given by the expression 8+2.5f. How much will it cost to make an arrangement with 8 flowers? 3. Jen was the leading scorer on her soccer team. She scored 120 goals and had 20 assists in her high school career. a. The number n of points awarded for goals is given by 2g where g is the number of goals scores. How many points did Jen earn for goals? b. The point total is given by n+a where a is the number of assists. Use your answer from part (a) to find Jen s point total.

22 ALGEBRAIC PROPERTIES Mnemonic: CADIC Idea (-7) = (3 9) = (4 3) (x + 6) = 7x = = Given: x = y y = z Then: x = z = = = (5 + 6) + 7 = 5 + (6 + 7) 11. (6 3) 4 = 4 (6 3) = 1