Ch 5 Notes

Algebra I Chapter 5 Mrs. Cataldo Class Notes Chapter 5.1 -5.8 SHORT QUIZ 1. Circle the numbers that are rational. a. 7 b. 0.25 c. 1/3 d. π 2. Round to the nearest 100th. a. 0.0017 b. 3.995 c. – 1.623 3. Solve a. -6.3 --7.01 = b. -2.8(0.4) = c. -2.8/4 = 5.1 Equations An equation is a math sentence that contains an ____________________. Equations can be ____________ or __________________. Example 6 --1 = 7 is ____________ but 6 --1 = 5 is _____________. Some equations are true or false depending on the value of a variable. For which value of x is this equation true? -9x = -81 x = ? In this case 9 is a _____________________ of the equation. A number that can be used to replace a variable in a an equation to make it true is called a __________________________ of the equation. An equation can have more than one solution or no solutions. What are the solutions for the following equations? 1. x² = 81 x= 2. x = x – 3 x = 5.2 Inverse Operations Inverse operations are operations that ______________another one. The inverse of multiplication is ____________. x • y = x Any number multiplied by another number and then divided by the y same number is the original number. P • 9 = P Note: the multiplicative inverse of 9 is 1/9 9 because 9 • 1/9 = 1, so 9x • 1/9 = x Also the inverse of division is _________________. x/y • y = x or x/5 • 5 = x The inverse of addition is _________________________. r + 5 -5 = r You can undo an addition by subtracting the same number. The additive inverse of 5 is -5 because 5 + -5 = 0. Also the inverse of subtraction is __________________. x – y + y = x or written as an addition equation, x + -y + y = x Example: What operations on 7y + 2 would give y? First undo the +2 by subtracting 2 or adding -2. 7y + 2 + -2 = 7y Then divide 7y by 7 or multiply 7y by 1/7. 7y/7 = y or 7y (1/7) = y Examples: Write an expression, equation or number for the following. a) The number that must be added to 3 to give 10. Let y be the number we are looking for. y+ 3 = 10 so y= 7 b) The number that must be added to 3 to give x. Let y be the number we are looking for. y + 3 = x so y = x + -3 5.3 Equivalent Equations Solving equations can be thought of as a balanced scale when the quantities on either side of the scale are the quantities on either side of the equals sign. For example the equation 2x + 3 = 9 can be represented by: Where x is an unknown weight represented by a box. x x = x In algebra, we can figure out how much each x box weighs by performing equivalent operations to each side of the scale or equals sign that will keep the scales balanced. If we take 3 balls off one both sides, will the scales balance? ______ We can represent this by subtracting 3 from both sides. Notice that we are performing the inverse operation for +3. 2x + 3 – 3 = 9 + -3 2x = 6 is an equivalent equation to the original 2x + 3 = 9 Again we can divide the quantities on both sides of the scale by 2. Will the scales balance? _______ This can be expressed mathematically by dividing both sides by 2. x x = x x = x 2x = 6 2 2 or x = 3 which is also an equivalent equation to the original 2x + 3 = 9 and tells us the weight of the box, x which is 3. We can check to see if 3 is a solution to the equation by plugging it in to see if the equation is true. 2(3) + 3 = 6 + 3 = 9 √ This is true. Example: Solve the following for x. Show your work. x + 12 = 3 4x = -16 x/-2 = -3.1 x/4 -2 = 10 5.4 Equivalent Expressions Two algebraic expressions are _______________________ if they are equal for all values of their variables. Write examples for each property Commutative Property of Addition Associative Property of Addition Commutative Property of Multiplication Associative Property of Multiplication Distributive Property of Multiplication over Addition and Subtraction Distributive Property of Division over Addition and Subtraction Examples: Is 3(4x) = 4(3x)? What property is this?__________________________ Is 4y + 2y = 2y + 4y ? What property is this? ______________________ Is 6(2p + 3) = 12p + 18? What property is this? ________________________ Simplify (10x – 6) 2 By the distributive rule you divide all terms by 2. So 10x/2 + -6/2 = 5x + -3 Also when you simplify, you can combine like terms. REVIEW Simplify 6 – (x + 8) When a subtraction sign appears outside of parentheses, think about the problem as follows. First add the opposite and remember that that the minus sign before the parenthesis is actually -1. 6 + -1(x + 8) You have to distribute the -1 over both terms in the parentheses. 6 + (-1)x + (-1)8 6 + -x + -8 -2 + -x 5.5 More on Solving Equations Strategy – combine like terms using addition or subtraction then divide or multiply in order to get x by itself on one side of the equation. Which terms are alike? Put a box around one set of terms that can be combined and a triangle around the other. First use the scale to illustrate 5x + 1 = 2x + 7 1. Subtract 2x from both sides of the equation. 5x – 2x +1 = 2x -2x + 7 3x + 1 = 7 2. Subtract 1 from both sides of the equation. 3x + 1 -1 = 7 – 1 3x = 6 3. Divide both sides by 3. = x = 2 Check 5(2) + 1 = 10 + 1 = 11 2(2) + 7 = 4 + 7 = 11 √ Why do we combine the like terms by addition or subtraction before we use multiplication or division? Another example: Solve for x 4(x -11) = 6 -x This time we see an expression in parentheses. Usually it is a good strategy to do any operations necessary to remove them first. 1. Use the distributive rule to simplify the left side. 4(x) -4(11) 4x + -44 We get 4x + -44 = 6 + -x 2. Combine like terms. Add x to both sides of the equation. 4x + -44 = 6 + -x +x +x 5x + -44= 6 3. Add 44 to both sides of the equation. 5x + -44 = 6 +44 +44 5x = 50 5. Get one x on one side of the equation. Divide by 5. = x = 10 6 Check 4(10 – 11) = 4(-1) = -4 6 – (10) = -4 √ Problem 5-15 GATES PROBLEMS – In class problem, copy from the board. 5.6 Length and Area Perimeter: The sum of the _______________________ of its sides. The units are length units. Area: The _____________________ of the length times the width of a rectangle. The units are squared. Example: The length is 3 ft, width is 2 ft. The perimeter is 3ft + 3ft + 2ft + 2ft= 10ft The area is 3ft • 2 ft = 6 ft² Where 6, 1ft by 1ft squares will fit inside the rectangle. When a dimension of a rectangle is unknown we can solve for it if we know the total area or perimeter and the length of one side. Example: Find the dimensions of this rectangle given that the perimeter is 22 inches. 2 ft 2 ft 3 ft 3 ft x-3 ft x-3 ft x ft x ft Solve for x first. We can write the equation x + x + (x-3) + (x-3) = 22 inches First combine like terms. 4x + -6 = 22 Then add 6 to both sides. 4x + -6 +6 = 22 + 6 4x = 28 Divide by 4 to get x alone on one side. x/4 = 28/4 x = 7 inches So one side is 7 inches long and the other is 7 – 3 inches or 4 inches long Check: 7 + 7 + 7 + -3 + 7 + -3 = 22 √ Example: Problem 9d p. 217 Find the length of each segment in each of the following diagrams if AB and CD are the same length. Since AB and CD are equal we can write this equation: x + 6(x+2) = 2(x + 10) A B E C D x 6(x+2) 2(x+10) Now we want to solve for x. Then we can calculate the lengths of each segment. First, simplify by using the ___________________rule. x + 6(x) + 6(2) = 2(x) + 2(10) x + 6x +12 = 2x + 20 Next combine like terms. 7x + 12 = 2x + 20 -2x -2x Subtract 2x from both sides. 5x + 12 = 20 -12 -12 Subtract 12 from both sides. 5x = 8 5 5 Divide both sides by 5 x = 1.6 units Substitute 1.6 for x in each segment length. AE = x = 1.6 units EB = 6(1.6 + 2) = 3.6 x 6 = 21.6 units AC = 2(1.6 + 10) = 2 x 11.6 = 23.2 units Check 1.6 + 21.6 = 23.2 √ More examples: 5.7 Distance, Rate and Time The distance traveled by something moving at a constant speed is the product of the ________________ of speed and the _______________ traveled. This relationship is expressed by the formula, d = rt Example: 5.8 Rate Problems – Example problem p. 228 #5 in virtual class