Math 8 Notes Units 1B: One-Step Equations and Inequalities Solving Equations Syllabus Objective: (1.10) The student will use order of operations to solve equations in the real number system. Equation a mathematical sentence that uses an equal sign to show that two expressions have the same value Inverse Operations operations that undo each other Solving Equations finding the value(s) of x which make the equation a true statement Strategy for Solving Equations: To solve linear equations, put the variable terms on one side of the equal sign, and put the constant (number) terms on the other side. To do this, use OPPOSITE (or INVERSE) OPERATIONS. Let s look at a gift wrapping analogy to better understand this strategy. When a present is wrapped it is placed in a box, the cover is put on, the box is wrapped in paper, and finally a ribbon is added to complete the project. To get the present out of the box, everything would be done in reverse order, performing the OPPOSITE (INVERSE) OPERATION. First we take off the ribbon, then take off the paper, next take the cover off, and finally take the present out of the box. Solving Equations Model Manipulatives can be used to model how to solve an equation. Let s take a look. KEY Remember = +1 = 1 It will not change the value of an expression is you add or remove zero. + = 0 Holt: Chapter 1, Sections 7-9 Math 8, Unit 1B: One-Step Equations and Inequalities Page 1 of 9
Example: Solve x 5 10. To solve the equation, you need to get x alone on one side of the equal sign. You can add or remove tiles as long as you add the same amount or remove the same amount on both sides. x 5 10 Since we are trying to solve for x, let s remove 5 the following: from each side. That will leave us with Remove 5 from each side. Remove 5 from each side. x = 5 Holt: Chapter 1, Sections 7-9 Math 8, Unit 1B: One-Step Equations and Inequalities Page 2 of 9
Example: Solve x 4 2. x + 4 = 2 In this example we need to remove 4 from both sides, but we don t have enough on the right side. Therefore, we must add in some zero pairs in order to have enough to take away. Now we have enough to remove 4 from both sides. Remove 4 from both sides. x = 2 Provide your students will plenty of practice using manipulatives. Once the students are comfortable using the manipulatives, ask them to start drawing pictures instead. Finally, once the students have mastered drawing the equations, you can move to the abstract. Holt: Chapter 1, Sections 7-9 Math 8, Unit 1B: One-Step Equations and Inequalities Page 3 of 9
To solve equations in the form of x b c, we will undo this algebraic expression to isolate the variable. To accomplish this, we will use the opposite operation to isolate the variable. Let s start with the Addition Property of Equality. ADDITION PROPERTY OF EQUALITY Words Numbers Algebra You can add the same number to both sides of an equation, and the statement will still be true. Example: Solve for x, x 5 8. x 5 8 5 5 x 13 x 5 8 13 5 8 8 8 Isolate the variable by using the Addition Property of Equality. Check the solution by using substitution: substitute your answer of 13 for x. Now let s look at the Subtraction Property of Equality. SUBTRACTION PROPERTY OF EQUALITY Words Numbers Algebra You can subtract the same number from both sides of an equation, and the statement will still be true. Example: Solve for t,. 6 t 28 6 6 t 22 6 t 28 6 22 28 28 28 Isolate the variable by using the Subtraction Property of Equality. Check the solution by using addition. Holt: Chapter 1, Sections 7-9 Math 8, Unit 1B: One-Step Equations and Inequalities Page 4 of 9
More examples: m 8 14 8 8 m 22 Check: m 8 14 22 8 14 14 14 15 w ( 14) 14 14 29 w Check: 15 w ( 14) 15 29 ( 14) 15 15 Manipulatives can be used to model how to solve equations using division. It is recommended that you use integers that divide evenly. Example: Solve 3n 9. 3n = 9 In this equation we need to isolate n. If we can split 9 up evenly among the n pieces then we will know what n equals. Each n has a value of 3. We know our answer then must be n = 3. Ask your students to identify the operation that was performed to solve for n. You may need to model a few examples before they are able to answer with confidence. Next, let s look at the Division Property of Equality. Holt: Chapter 1, Sections 7-9 Math 8, Unit 1B: One-Step Equations and Inequalities Page 5 of 9
DIVISION PROPERTY OF EQUALITY Words Numbers Algebra You can divide both sides of an equation by the same 4 3 12 nonzero number, and the 2 2 statement will still be true. 6 6 Example: Solve for x, 8x 32. 8x 32 8x 32 8 8 x 4 Isolate the variable by using the Division Property of Equality. 8x 32 8(4) 32 32 32 Check. Finally, let s look at the Multiplication Property of Equality. MULTIPLICATION PROPERTY OF EQUALITY Words Numbers Algebra You can multiply both sides of 2 6 an equation by the same 4 (2) 4 (6) numbers, and the statement will still be true. 24 24 Example: Solve for h, h 6. h 6 h 6 h 18 Isolate the variable by using the Multiplication Property of Equality. h 6 18 6 Check. 6 6 Holt: Chapter 1, Sections 7-9 Math 8, Unit 1B: One-Step Equations and Inequalities Page 6 of 9
More examples: 9y 45 9y 45 9 9 y 5 b 5 4 b 4 4 5 4 b 20 Check. 9y 45 9( 5) 45 45 45 b 5 4 20 5 4 Inequalities Syllabus Objective: (1.11) The students will solve inequalities. Syllabus Objective: (1.12) The student will graph inequalities with integers. We use inequalities in real life all the time. If you are going to purchase a $2 candy bar, you do not have to use exact change. How would you list all the amounts of money that are enough to buy the item? You might start a list: $3, $4, $5, $10; quickly you would discover that you could not list all possibilities. However, you could make a statement like any amount of money $2 or more and that would describe all the values. In algebra, we use inequality symbols to compare quantities when they are not equal, or compare quantities that may or may not be equal. symbol meaning words to look for < is less than below, fewer than, less than > is greater than above, must exceed, more than is less than or equal to at most, cannot exceed is greater than or equal to at least, no less than The solution of an inequality with a variable is the set of all numbers that make the statement true. You can show this solution by graphing on a number line. Holt: Chapter 1, Sections 7-9 Math 8, Unit 1B: One-Step Equations and Inequalities Page 7 of 9
inequality words graph All numbers less than 2. All numbers greater than 1. All numbers less than or equal to -1. All numbers greater than or equal to -2. Note that an open circle is used in the is less than or is greater than graphs, indicating that the number is not included in the solution. A closed circle is used in the is greater than or equal to or is less than or equal to graphs to indicate that the number is included in the solution. We can solve linear inequalities the same way we solve linear equations. We use the Order of Operations in reverse, using the opposite operation. Linear inequalities look like linear equations with the exception they have an inequality symbol (,, >, or < ) rather than an equal sign. At this point in our study, we will only address one-step inequalities with addition or subtraction. Linear Equation x 6 10 6 6 x 4 Linear Inequality x 6 10 6 6 x 4 Holt: Chapter 1, Sections 7-9 Math 8, Unit 1B: One-Step Equations and Inequalities Page 8 of 9
x 4 7 4 4 x x 4 7 4 4 x The following are websites that offer tiles and powerpoints that you may find useful. http://www.jamesrahn.com/algebra/pages/addition_of_integers.htm http://mathbits.com/mathbits/algebratiles/algebratilesmathbitsnew07impfree.html Holt: Chapter 1, Sections 7-9 Math 8, Unit 1B: One-Step Equations and Inequalities Page 9 of 9