Multi-Step Equations and Inequalities Syllabus Objective (1.13): The student will combine like terms in an epression when simplifying variable epressions. Term: the parts of an epression that are either added or subtracted. Like term: terms that have the same variable raised to the same power. Equivalent epression: epressions that have the same value for all values of the variables. Simplify: to write an epression in simplest form by performing all possible operations, including combing like terms. Combining Like Terms Algebra tiles can be used to teach student the concept of combining like terms. Eample: Combine like terms. A. 3+ 6 1. 1. This represents the original epression. 2. 2. This shows how the tiles could be moved to represent the combining of the like terms. 3. 3. The final answer: 3 Holt: Chapter 11, Sections 1- Math 8, Unit 11, Multi-Step Equations and Inequalities Page 1 of 11
Eample: Combine like terms. A. 1a a 9a 1. identify like terms. 2. combine coefficients: 1 = 9 B. 7y+ 8 3y 1+ y y + 7 Combining Like Terms in Two-Variable Epressions Eample: Combine like terms. A. 9+ 3y 2+ 9 2+ 3y+ 7+ 3y+ 1. Identify like terms. In this eample I rearranged the terms so that the like terms were net to each other. This would be a great time to review the commutative property. 2. Combine the coefficients. B. m+ 9n 2 There are no like terms in this eample. This epression is already written in simplest form. Using the Distributive Property to Simplify Eample: Simplify A. 6( + n) 6() + 6( n) 30 + 6n n + 30 Remember: the Distributive Property states that a(b+c) = ab + ac for all real numbers a, b, and c. Multiply. Combine coefficients. Holt: Chapter 11, Sections 1- Math 8, Unit 11, Multi-Step Equations and Inequalities Page 2 of 11
Solving Multi-Step Equations Syllabus Objective (9.1): The student will solve two-step equations. Syllabus Objective (9.2): The student will solve multi-step equations. The general strategy for solving a multi-step equation in one variable is to rewrite the equation in a + b = c format, then solve the equation by isolating the variable using the Order of Operations in reverse and using the opposite operation. (Remember the analogy to unwrapping a gift ) Order of Operations Parentheses (grouping symbols) Eponents Multiply/Divide from left to right Add/Subtract from left to right Evaluating an arithmetic epression using the Order of Operations will suggest how we might go about solving equation in the a + b = c format. To evaluate an arithmetic epression such as + 2, we would use the Order of Operations. + 2 + 10 1 1. multiply, 2 2. add, + 10 Now, rewriting that epression, we have 2 + = 1, a form that leads to equations written in the form a + b = c. If I replace with n, I have: 2 n + = 1 or + = 1 an equation in the a + b = c format. To solve that equation, I am going to undo the epression +. I will isolate the variable by using the Order of Operations in reverse and using the opposite operation. That translates to getting rid of any addition or subtraction first, then getting rid of any multiplication or division net. Undoing the epression and isolating the variable results in finding the value of n. Holt: Chapter 11, Sections 1- Math 8, Unit 11, Multi-Step Equations and Inequalities Page 3 of 11
This is what it looks like: + = 1 + = 1 = 10 10 = 2 2 n = subtract from each side to undo addition divide by 2 to undo the multiplication + = 1 = = 10 10 = 2 2 n = Check your solution by substituting the answer back into the original equation. + = 1 2() + = 1 10 + = 1 1 = 1 original equation substitute for n true statement, so the solution is correct Eample: Solve for, = 17. Check the solution: = 17 3(7) = 17 21 = 17 17 = 17 = 17 + = + = 21 21 = 3 3 = 7 OR = 17 + = 17+ = 21 21 = 3 3 = 7 Holt: Chapter 11, Sections 1- Math 8, Unit 11, Multi-Step Equations and Inequalities Page of 11
Eample: Solve for, + = 12. + = 12 + = 12 = 7 ()( ) = ()(7) = 28 Check: OR + = 12 + = 12 = = 7 ()( ) = ()(7) = 28 + = 12 (28) + = 12 7+ = 12 12 = 12 Note: Knowing how to solve equations in the a + b = c format is etremely important for success in algebra. All other equations will be solved by converting equations to a + b = c. To solve systems of equations, we first rewrite the equations into the a + b = c form and then solve. In the student s algebraic future, to solve quadratic equations we will rewrite the equation into factors using a + b = c, then solve the resulting equation letting c = 0. It is important that students are comfortable solving equations in the a + b = c format. CCSS 8.EE.7b: Solve linear equations with rational number coefficients, including equations whose solutions require epanding epressions using the distributive property and collecting like terms. Eample: Solve for, ( + 2) 3 = 3( 1) 2. In this problem, there are parentheses and variables on both sides of the equation. It is clearly a longer problem. The strategy remains the same: rewrite the equation in a + b = c format, and then isolate the variable by using the Order of Operations in reverse using the opposite operation. Physically, this equation looks different because there are parentheses, so let s get rid of them by using the distributive property. Holt: Chapter 11, Sections 1- Math 8, Unit 11, Multi-Step Equations and Inequalities Page of 11
( + 2) 3 = 3( 1) 2 + 10 3 = 3 2 Distributive Property + 7 = 3 Combine like terms + 7 = 3 Subtraction Property of Equality + 7 = 3 Combine like terms + 7 7 = 3 7 Addition Property of Equality = 10 Simplify 10 = Division Property of Equality = or 2. Simplify 2 Caution! When solving equations, we often write arithmetic epressions such as 3 + ( 7) as 3 7as we did in the above problem. Students need to be reminded that when there is not an etra minus sign, the computation is understood to be an addition problem. Eamples: ( + ) + ( + 6) = + 6= 11 ( ) + ( 6) = 6= 11 ( 8) + ( + 3) = 8+ 3= Emphasizing this will help students to solve equations correctly. If not taught, students often solve the equation correctly only to make an arithmetic mistake then they think the problem they are encountering is algebraic in nature (instead of arithmetic). Also make sure students understand simplifying epressions with several negatives. Eamples: ( ) = + = 9 3 ( 8) = 3 + 8 = 2 = 2 + ( ) = 7 Solving Equations with No Solutions or Infinitely Many as a Solution CCSS 8.EE.7a-1: Give eamples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. CCSS 8.EE.7a-2: Show whether a linear equation in one variable has one solution, infinitely many solutions or no solutions by successively transforming the given equation into simpler forms, until an equivalent equations of the form = a, a= a, or a= b results (where a and b are different numbers). Holt: Chapter 11, Sections 1- Math 8, Unit 11, Multi-Step Equations and Inequalities Page 6 of 11
When you solve an equation, you may find something unusual happens: when you subtract the variable from both sides (in an effort to get the variable term on one side of the equation), no variable term remains! Your solution is either no solution or infinitely many. Look at the following eamples. Eample: Solve ( ) ( ) 33 1= 9. 33 1= 9 9 3= 9 9 3= 9 9 = 9 3= 0 Notice we could stop now if we recognized that it is impossible for a number 9 to be equal to 3 less than itself. If we continue...we still have a statement that is not true. So the equation is said to have NO SOLUTION. Eample: Solve 8 2= 2( 1. ) 8 2= 2( 1) 8 2= 8 2 8 2= 8 2 8 = 8 2= 2 Notice that this statement would be true no matter what value we substitute for. We could go further.. We still get a true statement. So the equation is said to have a solution of INFINITELY MANY. Inequalities Syllabus objective (2.9): The student will solve inequalities with rational numbers. Syllabus objective (9.3): The student will solve two-step inequalities. Syllabus objective (9.): The student will graph the solution to two-step inequalities on a number line. Holt: Chapter 11, Sections 1- Math 8, Unit 11, Multi-Step Equations and Inequalities Page 7 of 11
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 eact change. How would you list all the amounts of money that are enough to buy the item? You might start a list: $3, $, $, $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 Disguised in word problems as < is less than below, fewer than, less than > is greater than above, must eceed, more than is less than or equal to at most, cannot eceed, no more than 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. Inequality In words Graph < 2 all numbers less than two > 1 all numbers greater than one 1 all numbers less than or equal to negative one 2 all numbers greater than or equal to negative two Note: 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. Holt: Chapter 11, Sections 1- Math 8, Unit 11, Multi-Step Equations and Inequalities Page 8 of 11
Reading an Inequality We know > means greater than when we read from left to right. So > is read five is greater than. When I read from right to left, the way the inequality is read changes, and it also means four is less than five. This could be written <. In the problems that follow, we may have a solution like >. Reading from left to right, we say negative four is greater than. Generally in algebra, the convention is to always read the variable first. So, rather than saying negative four is greater than, we would say is less than negative four and rewrite it as <. That suggests these statements are equivalent: > < Eample: Which graph represents the solution set of the inequality 2 >? (Note that a problem similar to this appeared in the CRT Instructional Materials.) A B C D 0 0 0 0 It is a common error for students to now select C as the correct answer. You may want to encourage students to rewrite the inequality with the variable on the left side, as < 2, which more clearly shows the answer of A. Solving an Inequality 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 eception that they have an inequality symbol rather than an equal sign. Holt: Chapter 11, Sections 1- Math 8, Unit 11, Multi-Step Equations and Inequalities Page 9 of 11
Eample: 2 = 10 2+ 2= 10+ 2 = 12 12 = 3 3 = 2 > 10 2+ 2> 10+ 2 > 12 12 > 3 3 > Notice the graph on the left only has the point representing. That translates to =. There is only one solution to the equation. The graph on the right has a dot on, which is not shaded because is not included as part of the solution. Also notice that there is a solid line to the right of the open dot, representing all the numbers greater than that are part of the solution set. But we need to consider what happens to the value of an inequality if we multiply or divide by a negative number. What would happen if I had an inequality such as 6> and multiplied both sides of the inequality by 2? 6 > This is a true statement. ( 2)6 > ( 2)( ) Is this true? 12 > 10 This is NOT true. If we look at the number line, we notice that 12 would be to the left of 10 on the number line. Numbers to the left are smaller, so 12 < 10. It appears when we multiply an inequality by a negative number, to make the statement true, the order of the inequality must be reversed. Note the inequality below. When both sides are divided by a negative number, the order of the inequality must also be reversed. 10 > 1 This is true. 10 1 > Is this true? 2 > 3 This is NOT true! What is true: 2< 3; we had to reverse the inequality to make a true statement. Multiplication Property of Inequality If a < b and c > 0, then ac < bc. If a < b and c < 0, then ac > bc. In other words, when you multiply an inequality by a negative number (c < 0), the direction of the inequality changes. Holt: Chapter 11, Sections 1- Math 8, Unit 11, Multi-Step Equations and Inequalities Page 10 of 11
a b Division Property of Inequality If a < b and c > 0, then <. c c a b If a < b and c < 0, then >. c c That is, when you divide an inequality by a negative number, the direction of the inequality changes. Eample: Find the solution set for 2 > 10. 2 > 10 2+ 2> 10+ 2 > 12 12 < 3 3 < Check: Pick any number in the solution set and substitute it into the original inequality. 2 > 10 ( 3)( 10) 2 > 10 Eample: 30 2 > 10 28 > 10 Rule: when you multiply or divide an inequality by a negative number, you reverse the sign of the inequality. Remember, if you don t keep the coefficients of the variable positive, then you may have to multiply or divide by a negative number that will require you to change the direction of the inequality. Holt: Chapter 11, Sections 1- Math 8, Unit 11, Multi-Step Equations and Inequalities Page 11 of 11