By: Claudia Subtracting Integers Subtracting integers: 2 – (-8) Rule When we subtract we ADD THE OPPISITE! (never change the first number!) 2 + 8 is the same as 2 – (-8)….. So, the answer is the same, 10 -8 + 2 = -6 The answer is positive because you always use the sign of the number with the highest absolute value. -8 is farther away from zero then 2 is. So -8 has the highest absolute value. Vocab: Integers – the set of whole numbers and their opposites. Absolute Value – the distance the number is from zero on the number line. 2
How to Multiply Integers When multiplying integers with the same sign the product is always positive. When multiplying integers with different signs the answer will always be negative. If any of the integers is zero the result is always zero. 3
Ex: 5+3=8 Add the numbers as if they were positive, then add the sign of the numbers. Adding Integers having the same sign= Ex: -5+(-3)= -8 2. Adding two integers having signs: Ex: -5+3 Take the difference of the numbers as if they were positive, then add the sign of the number having absolute value. Ex: -5+3=-2 Subtracting Integers Adding Integers
=-5+3 When we subtract, we ADD THE OPPOSITE Rule: 2-1 then turns into 2+(-1) It helps us to get the right answer, too, and less confusing. 4
Multiplying and Dividing Integers Dividing Integers
If a pair of integers has the same sign, then the answer will have a positive sign. You must calculate the absolute value of each integer and then divide the first integer by the second integer.
Example: -10 / -2 = ?
Step 1: |-10| / |-2| = 10 / 2 Step 2: 10 / 2 = 5 Step 3: Since integers have same sign, answer is positive: +5
If a pair of integers have different signs, then the answer will be negative. You must calculate the absolute value of each integer and then divide the first integer by the second integer.
Example: -10 / +2 = ?
Step 1: |-10| / |+2| = 10 / 2 Step 2: 10 / 2 = 5 Step 3: Since integers have different signs, answer is negative: -5 Multiplying Integers
When multiplying two integers having the same sign, the product is always positive
Example 1: -2 · (-5) = 10 Example 2: 2 · 5 = 10
When having two integers with different signs, the product is always negative
Example 1: -2 · 5 = (-2)+(-2)+(-2)+(-2)+(-2) = -10 Example 2: 2 · (-5) = (-5)+(-5) = -10
When multiplying more than two integers
Example 1: (-1) · (-2) · (-3) = ? Step 1: group the first two numbers and use rules I and II above to calculate the intermediate step (-1) · (-2) = +2 (used rule I) Step 2: use result from intermediate step 1 and multiply by the third number. 2 · (-3) = -6 (used rule II)
AddingandSubtractingIntegers When adding Integers with the same sign add them as if they were positive then add the sign. Example: 6 + 3 = 9 -6 + (-3) = -9 When adding integers with different signs, subtract them as if they were positive and add the sign of the number with greatest absolute value. Example: -6 + 3 = -3 When subtracting any integer you add the opposite. Example: -6 – 3 = -9 Change to -6 + (-3) = -9 Hannah 6
Subtracting Integers Ex 126-(-176) 126+176=302 or 126-176 126+(-176)=(-50) or -126-(-176) -126+176=50 (note) when you add integers remember that when you add integers with the same sign the answer is going to be the same as the sign, but if the absolute value of the negative number ishigher than the positive thanthenumbers going to be anegative. Convert the problem to addition. Ex. 12-(-36) to 12+36. remember to change the last number of the sequence from negative to positive or positive negative. Add or subtract the problem like a regular math problem. Ex. 12+36=48. 7
Solving Equations When the number in the equation is positive you add the opposite to the number. Then you add the opposite to the answer. That way, the variable is alone on the left side of the equation in this example, and the difference of the answer and the opposite number is on the other. When the number in the equation is negative then you convert the number to a positive. Then you change the operation to its opposite. After that, you add the opposite to the number. Then you add the opposite to the answer. That way, the variable is alone on the one side of the equal sign, and the difference of the answer and the opposite number is on the other. Example X+13=26 X+13+(-13)=26+(-13) 26+(-13)=13 13+(-13)=0 X=13 13+13=26 Example X-(-13)=13 X+13+(-13)=26+(-13) 26+(-13)=13 13+(-13)=0 X=13 13-(-13)=26 8
Distributive Property For Algebra Take both numbers in the parentheses and multiply them separately to the number outside of the parentheses, still using the sign in between both numbers in parentheses. Ex. 1 : 5(Y+9) turns into 5y+59 = 5Y+45 Ex. 2 : 5(-Y+9) turns into 5(-Y)+59 = 5(- Y)+45 Ex. 3: -5(Y-9) turnsinto -5Y-(-59) = -5Y - (-45) 9
How to Solve Equations Created By: Jonah Step 1 A legal move (you have to do the same thing to both sides) is very simple. Step 2 Step 3 What you are trying to do here is; you want to get the variable alone. All you have to do is add the opposite to the constant Once the constant is gone, you add the same number you added to the sum, then whatever you get from that equation, is what the variable equals Example: X + 5 = 12 X + 5 + (-5) = 12 + (-5) Example: X + 5 + (-5) = 12 + (-5) X = 7 10
Subtracting Integers By: Cameron
When subtracting integers you “add the opposite”.
When Subtracting Integers you add the opposite. Example: 10-(-4)=14 10+4=14 Do you want to know how this works~ click to find out. 11
Just draw a number line if it helps you more. Also when you have a subtraction sign next to a parenthesis. You change the sign to addition and the negative number to a positive. Example2: -10-(4)=6 -10+-4=14 Example: -10-(-4)=14 10+4=14 12
HOW TO COMBINE LIKE TERMSA.K.A. SIMPLIFYING ALGEBRAIC EXPRESSIONS Congrats you can now simplify algebraic expressions!!!!!!!!
Terms --- The algebraic expression separated between each plus or minus sign
Ex. 3x, y, 2x, 7
Like terms --- Terms that conduct the same variables
Ex. 3x & 2x
Coefficients --- The numbers that are involved with a variable
Ex. 3, 2, 1
Constants --- Terms without an variable
Ex. 7 Step 1 Step 3 Hint The only like terms are 3x & 2x
You begin with an Algebraic Expression to simplify
3x+y+2x+7=? Step 2 3x+2x=5x Final answer 5x+y+7 The Surprise Expression and find…………………..
Before we simplify, find the terms, like terms, coefficients, and constants.
Solving an algebraic equation!!!! You solve an algebraic equation by doing different sets of legal moves. You do a legal move by adding or subtracting and in some cases multiplication and dividing what you do to one side to the other until you cant do anymore moves. example: 3+4+-4=Y+4+-4+3 By Lennon Dresnin
(-2)2 ≠ -22 Exponents Exponents This is where your journey into Exponents begins By Loghan In (-2)2 You can tell (-2) is the base because it is in parenthesis, In -22 there are no parenthesis. Because of that 2 is the base not -2. Another way of doing the problem would be 0-22. In 0-22 you would start off by doing 2 to the second power, which is 2 times 2. The answer would be 4. But after you do that the equation would be 0-4. 0-4=(-4). The answer is different than (-2)2. In (-2)2 you can tell the base is (-2) because there are parenthesis. There is no chance of it being -2 or minus two. Now all we need to find out is what negative 2 multiplied by itself is. When you multiply a negative by a negative what do you get? A positive! So when you multiply -1 by -1 you get 1! Positive 1. So when you multiply -2 by -2 [In other words (-2)2] you get… By Loghan I hope you enjoyed your journey into Exponents 22=(-2)2 22=-22 So this will sum everything up: (-2)2 = 4 Does it work? It Works OH NO IT DOESN’T WORK -22 = -4