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Pre-Algebra Notes – Unit Three: Multi-Step Equations and Inequalities (optional) CCSD Teachers note: CCSD syllabus objectives (2.8)The student will solve multi-step inequalities and (2.9)The student will graph solutions to multi-step inequalities have been eliminated for 2011-2012. However, we have chosen to include the notes as some teachers have expressed that they will still include the concepts in their classrooms. A note to substitute teachers: pre-algebra teachers agree that all units of study are important, but understanding this unit seems to be a key as to whether or not a student is successful with the following chapters. While it is important to maintain the pace of the CCSD benchmark calendar, veteran teachers warn us not to rush through this material. Solving Two-Step Equations Syllabus Objective: (2.4) The student will solve two-step equations. The general strategy for solving a multi-step equation in one variable is to rewrite the equation in ax + b = c format (where a, b, and c are real numbers), 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…) 1. 2. 3. 4. Order of Operations Parentheses (Grouping) Exponents Multiply/Divide, left to right Add/Subtract, left to right Evaluating an arithmetic expression using the Order of Operations will suggest how we might go about solving equations in the ax + b = c format. To evaluate an arithmetic expression such as 4 + 2 ⋅ 5 , we’d use the Order of Operations. = 4 + 2⋅5 = 4 + 10 14 First we multiply, 2 ⋅ 5 Second we add, 4 + 10 Now, rewriting that expression, we have 2 ⋅ 5 + 4 = 14 , a form that leads to equations written in the form ax + b = c. If I replace 5 with n, I have 2⋅n + 4 = 14 or 2n + 4 = 14, an equation in the ax + b = c format. McDougal Littell, Chapter 3 HS Pre-Algebra , Unit 03: Multi-Step Equations and Inequalities Revised 2013 - CCSS Page 1 of 14 To solve that equation, I am going to “undo” the expression “2n + 4”. 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 next. Undoing the expression and isolating the variable results in finding the value of n. This is what it looks like: 2n + 4 14 2n + 4 14 = 2n + 4 − 4 = 14 − 4 subtract 4 from each side to "undo" the addition − 4 =−4 2n 10 2n 10 = 2n 10 2n 10 divide by 2 to "undo" the mulitplication = = 2 2 2 2 n 5= n 5 Check your solution by substituting the answer back into the original equation. 2n + 4 = 14 2(5) + 4 = 14 10 + 4 = 14 14 = 14 Example: original equation substitute '5' for 'n' true statement, so my solution is correct Solve for x, 3x − 4 = 17 . Using the general strategy, we always want to “undo” whatever has been done in reverse order. We will undo the subtracting first by adding, and then undo the multiplication by dividing. = 3x − 4 17 = or 3x − 4 17 3x − 4 + 4 = 17 + 4 + 4 =+4 = 3x 3x = 3 = x Check: 21 = 3x 21 21 3x 21 = 3 3 3 7= x 7 3x − 4 = 17 3(7) − 4 = 17 21 − 4 = 17 17 = 17 McDougal Littell, Chapter 3 HS Pre-Algebra , Unit 03: Multi-Step Equations and Inequalities Revised 2013 - CCSS Page 2 of 14 Example: Solve for x, x + 5 = 12 4 x x = + 5 12 = or + 5 12 4 4 x + 5 − 5 =12 − 5 − 5 =−5 4 x x = 7= 7 4 4 x x (4) = (4)(7) (4) = (4)(7) 4 4 x 28 x 28 = = Check: x +5= 12 4 (28) +5= 12 4 7+5= 12 12 = 12 ***NOTE: Knowing how to solve equations in the ax + b = c format is extremely important for success in algebra. All other equations will be solved by converting equations to ax + b = c . To solve systems of equations, we rewrite the equations into one equation in the ax + b = c form and solve. (In the student’s algebraic future, to solve quadratic equations we will rewrite the equation into factors using ax + b = c , then solve the resulting equation letting c = 0 . It is important that students are comfortable solving equations in the ax + b = c format.) Solving Equations that Are NOT in the ax + b = c Format Syllabus Objective: (2.5) The student will solve equations with like terms and parentheses. The general strategy for solving equations NOT in the ax + b = c format is to rewrite the equation in ax + b = c format using the Properties of Real Numbers. An important problem solving/learning strategy is to take a problem that you don’t know how to do and transform that into a problem that you know how to solve. Right now, we know how to solve problems such as ax + b = c . I cannot make that problem more difficult, but what I can do is make it look different. For instance, if I asked you to solve 5(2 x + 3) − 4 = 21 , that problem looks different from the 2x + 4 = 14 that we just solved by using the Order of Operations in reverse. The question you have to ask is “what is physically different in these two problems?” McDougal Littell, Chapter 3 HS Pre-Algebra , Unit 03: Multi-Step Equations and Inequalities Revised 2013 - CCSS Page 3 of 14 The answer is the new problem has parentheses. That means to make these problems look alike, we need to get rid of those parentheses. We’ll do that by using the distributive property. 5(2 x + 3) − 4 = 21 10 x + 15 − 4 = 21 10 x + 11 = 21 we applied the distributive property we combined like terms Now I have converted 5(2 x + 3) − 4 = c format as 10 x + 11 = 21 . We recognize 21 into ax + b = this format and know how to solve it. = = or 10 x + 11 21 10 x + 11 21 − 11 = −11 10 x + 11 − 11 = 21 − 11 = = 10 x 10 10 x 10 10 x 10 10 x 10 = = 10 10 10 10 = x 1= x 1 Check: 5(2 x + 3) − 4 = 21 5 [ 2(1) + 3] − 4 = 21 5 [ 2 + 3] − 4 = 21 5 [5] − 4 = 21 25 − 4 = 21 21 = 21 Example: Solve for x, 4(3x − 2) + 5 = 45 . The parentheses make this problem look different than the ax + b = c problems. Get rid of the parentheses by using the distributive property, and then combine terms. 4(3x − 2) + 5 = 45 12 x − 8 + 5 = 45 12 x − 3 = 45 12 x − 3 + 3 = 45 + 3 12 x = 48 12 x 48 = 12 12 x=4 We will leave the check to you! McDougal Littell, Chapter 3 HS Pre-Algebra , Unit 03: Multi-Step Equations and Inequalities Revised 2013 - CCSS Page 4 of 14 Caution! Note that when we solved these equations we got rid of the parentheses first. Students will often want to get rid of the addition/subtraction first because they are using the Order of Operations in reverse. The reason we get rid of the parentheses first is because the strategy is to rewrite the equation in ax + b = c format before using the Order of Operations in reverse. Reading an Inequality (optional; see note on page 1) We know “>” means “greater than” when we read from left to right. So 5 > 4 is read “five is greater than 4”. 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 4 < 5. In the problems that follow, we may have a solution like −4 > x . Reading from left to right, we say “negative four is greater than x”. Generally in algebra, the convention is to always read the variable first. So, rather than saying “negative four is greater than x”, we would say “x is less than negative four” and rewrite it as “ x < −4 . That suggests these statements are equivalent: −4 > x ⇔ x < −4 Example: Which graph represents the solution set of the inequality −2 > x ? (Note that a problem similar to this appeared in the CRT Instructional Materials.) A B C D −5 0 5 −5 0 5 −5 0 5 −5 0 5 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 x < −2 , which more clearly shows the answer of “A”. McDougal Littell, Chapter 3 HS Pre-Algebra , Unit 03: Multi-Step Equations and Inequalities Revised 2013 - CCSS Page 5 of 14 Solving Linear Equations with Variables on Both Sides Syllabus Objective: (2.6) The student will solve equations with variables on both sides. Using the same strategy, we rewrite the equation so it is in ax + b = c format. Example: Solve for x, 5 x + 2 = 3x + 14 . We ask ourselves, “what is physically different in this problem?” This equation is different because there are variables on both sides of the equation. The strategy is to rewrite the equation in ax + b = c format which calls for variables on only one side of the equation. What’s different? There is a 3x on the other side of the equation. How do we get rid of the addition of 3x? 5x + 2 = 3x + 14 5x + 2 – 3x = 3x + 14 – 3x 5x – 3x + 2 = 3x – 3x + 14 2x + 2 = 14 Subtraction Property of Equality Commutative Property of Addition Combining like terms Now the equation is in ax + b = c format. 2x + 2 = 14 2x + 2 – 2 = 14 – 2 Subtraction Property of Equality 2x = 12 Arithmetic fact 2x 12 Division Property of Equality = 2 2 x=6 Arithmetic fact Let’s try a few more examples: 5 x − 17 = 3x − 9 −3x = −3 x 2 x − 17 = −9 +17 = +17 2x = 8 2x 8 = 2 2 x=4 −2 x − 17 =6 − x +x = +x −1x − 17 = 6 +17 = +17 −1 x = 23 −1x 23 = −1 −1 x = −23 3m + 8 − 5m = 9 + 4m + 29 −2m + 8 = 4m + 38 −2m − 4m + 8 = 4m − 4m + 38 −6 m + 8 = 38 −8 =−8 −6 m = 30 −6m 30 = −6 −6 m = −5 Now, I can make equations longer, but I cannot make them more difficult! McDougal Littell, Chapter 3 HS Pre-Algebra , Unit 03: Multi-Step Equations and Inequalities Revised 2013 - CCSS Page 6 of 14 CCSS 8.EE.7b: Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Example: Solve for x, 5( x + 2) − 3= 3( x − 1) − 2 x . 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 ax + 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. 5( x + 2) − 3= 3( x − 1) − 2 x 5 x + 10 − 3 = 3x − 3 − 2 x 5x + 7 = x − 3 5x + 7 − x = x − 3 − x 4 x + 7 =−3 4 x + 7 − 7 =−3 − 7 4 x = −10 4 x −10 = 4 4 −5 = x or − 2.5 2 Distributive Property Combine like terms Subtraction Property of Equality Combine like terms Addition Property of Equality Simplify Division Property of Equality Simplify Caution! When solving equations, we often write arithmetic expressions such as −3 + ( −7) as −3 − 7 as we did in the above problem. Students need to be reminded that when there is not an “extra” minus sign, the computation is understood to be an addition problem. Examples: ( +5) + ( +6) = 5 + 6 = 11 ( −5) + ( −6) =−5 − 6 =−11 ( −8) + ( +3) =−8 + 3 =−5 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 expressions with several negatives. Examples: 5 − ( −4) = 5 + 4 = 9 −3 − ( −8) =−3 + 8 =5 −2 − 5 =−2 + ( −5) =−7 McDougal Littell, Chapter 3 HS Pre-Algebra , Unit 03: Multi-Step Equations and Inequalities Revised 2013 - CCSS Page 7 of 14 Solving Equations with “No Solutions” or “Infinitely Many” as a Solution CCSS 8.EE.7a: Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. 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= x a= , a a, or = a b results (where a and b are different numbers). 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 examples. Example: Solve 3 ( 3x − 1) = 9x . 3 ( 3x − 1) = 9x 9x − 3 = 9x 9 x − 3 =9 x −9 x = −9 x −3 =0 Notice we could stop now if we recognized that it is impossible for a number 9x 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. Example: Solve 8 x −= 2 2 ( 4 x − 1) . 8 x −= 2 2 ( 4 x − 1) 8x − 2 = 8x − 2 8x − 2 = 8x − 2 Notice that this statement would be true no matter what value we substitute for x. We could go further…. −8 x = −8 x −2 =−2 McDougal Littell, Chapter 3 …. We still get a true statement. So the equation is said to have a solution of INFINITELY MANY. HS Pre-Algebra , Unit 03: Multi-Step Equations and Inequalities Revised 2013 - CCSS Page 8 of 14 Solving and Graphing Inequalities (optional; refer to note on page 1) Syllabus Objectives: (1.4) The student will solve inequalities involving integers. (2.7) The student will graph the solution set of a linear equation in one variable. (2.8) The student will solve multi-step inequalities. (2.9) The student will graph solutions to multi-step inequalities. We can solve linear inequalities the same way we solve linear equations. We write the inequalities in the ax + b > c format, then we use the Order of Operations in reverse using the opposite operation. Linear inequalities look like linear equations with the exception they have a greater than (>), greater than or equal to ( ≥ ), less than (<) or less than or equal to ( ≤ ) sign rather than just an equal sign. Examples: Linear Equation 3x – 2 = 10 Linear Inequality 3x – 2 > 10 Let’s solve each and graph the solution. Solving them both, we have: 3x − 2 10 = 3x − 2 + 2 = 10 + 2 3x 12 = 3x 12 = 3 3 = x 4 3x − 2 > 10 3x − 2 + 2 > 10 + 2 3x > 12 3x 12 > 3 3 x>4 To graph x = 4, we place a point on the number line at 4. −5 0 5 To graph x > 4, we will place an open circle at 4 because 4 is not included in the solution. We will “shade” or draw a solid line with an arrow to the right, which indicates that all the numbers greater than 4 are part of the solution set. −5 0 5 If the inequality contained an equal sign, 3x – 2 ≥ 10, then everything would be done the same in terms of solving the inequality, except the answer and graph would look a little different because it would include 4 as part of the solution set. To show 4 was included, we would shade it. The solution is x ≥ 4. −5 0 5 It may appear that the Properties of Equalities that we have studied will also work for Properties of Inequalities. They do for addition and subtraction, but for multiplication and division, they hold true for positive numbers only. McDougal Littell, Chapter 3 HS Pre-Algebra , Unit 03: Multi-Step Equations and Inequalities Revised 2013 - CCSS Page 9 of 14 Example: Find and graph the solution set for 3x − 12 ≤ 13. 3x − 2 ≤ 13 3x − 2 + 2 ≤ 13 + 2 3x ≤ 15 3x 15 ≤ 3 3 x≤5 −5 Addition Property of Inequality Combine like terms Division Property of Inequality Simplify 0 5 Again, I can’t make these problems more difficult, I can only make them longer as I did with linear equations. 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 > −5 and multiplied both sides of the inequality by –2? 6 > −5 ( −2)6 > ( −2)( −5) − 12 > 10 This is a true statement. Is this true? 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 > −15 10 −15 > −5 −5 −2 > 3 This is true. Is this true? 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. McDougal Littell, Chapter 3 HS Pre-Algebra , Unit 03: Multi-Step Equations and Inequalities Revised 2013 - CCSS Page 10 of 14 Division Property of Inequality a b < . c c a b If a < b and c < 0, then > . c c If a < b and c > 0, then That is, when you divide an inequality by a negative number, the direction of the inequality changes. Example: Find the solution set for −3x − 2 > 10 . − 3x − 2 > 10 −3x − 2 + 2 > 10 + 2 − 3x > 12 −3x 12 < −3 −3 x < −4 Notice that when we divided by a negative number, we reversed the order of the inequality. We could have done the same problem without multiplying or dividing by a negative number by keeping the variable positive. That would result in not having to reverse the inequality. Let’s look at the same problem. Example: Find the solution set for −3x − 2 > 10 . Rather than having the variables on the left side and have the coefficient negative, I could put the variables on the right side (add 3x to both sides) that would result in a positive coefficient. − 3x − 2 > 10 3x − 3x − 2 > 3x + 10 − 2 > 3x + 10 − 2 − 10 > 3x + 10 − 10 − 12 > 3x −12 3x > 3 3 or −4> x x < −4 * McDougal Littell, Chapter 3 HS Pre-Algebra , Unit 03: Multi-Step Equations and Inequalities Revised 2013 - CCSS Page 11 of 14 In both cases, we have the same answer. (*An explanation is in a preceding section Reading an Inequality.) In the second example, since I kept my variables positive, I did not have to reverse the order of the inequality. Bottom line, you solve linear inequalities the same way you solve linear equations: write them in ax + b = c format, then use the Order of Operations in reverse using the opposite operation. Remember, if you don’t keep the coefficients of the variable positive, then you will likely have to multiply or divide by a negative number that will require you to change the order of the inequality. Solving and Graphing Equations and Inequalities with Absolute Value CCSD Teachers note: This is not a CCSD syllabus objective, and the State Standard: (2.8.2)The student will solve and graphically represent equations and inequalities in one variable, including absolute value has been partially eliminated for 2011-2012. The Nevada State Grade 8 Transition Guide states “Continue to solve equations and inequalities in one variable. Graphing inequalities is in the CCSS in Grade 7 and High School.” The following material has been included for your general information. Some equations will involve absolute value. You must remember that two numbers can have the same absolute value. Example: Find and graph the solution set for x = 3. (A similar question was included on the CRT Instructional Materials.) Using the definition of absolute value, we think “what are the points that are each a distance of 3 from zero?” We know that there are 2 values for x that make this a true statement: x = 3 or x = −3. The graph would be: −5 0 5 Example: Find and graph the solutions set for x − 2 = 3. First, we must isolate the absolute value. 3 x −2= +2 =+2 x =5 McDougal Littell, Chapter 3 HS Pre-Algebra , Unit 03: Multi-Step Equations and Inequalities Revised 2013 - CCSS Page 12 of 14 We know that for x = 5 to be true, x = 5 or x = −5 because = 5 5 and = −5 5 . −5 5 0 Let’s check our solution by substituting the answers back into the original equation. x −2= 3 is the original equation = −5 − 2 3 = substitute answers 5 −2 3 = 5−2 3 = 5−2 3 3 = 3 3 = 3 Always remember to isolate the absolute value term before breaking the problem into two equations. 5. Example: Solve 2 x − 1 = First, let’s isolate the 2x . 2 x − 1 =5 +1 =+1 2x = 6 If 2 x = 6, then either 2x = −6 or 2x = 6 2 x −6 2x 6 = = 2 2 2 2 x= −3 or x= 3 −5 5 0 Let’s consider graphing simple absolute value inequalities. If we were asked to graph the solution to x < 2 , we can think “what are the values of x that are less than 2 units away from zero?” It would be the values between −2 and 2. −5 McDougal Littell, Chapter 3 0 5 HS Pre-Algebra , Unit 03: Multi-Step Equations and Inequalities Revised 2013 - CCSS Page 13 of 14 Likewise, let’s reason what the graph of x > 2 would look like. It would include the values of x that are more than 2 units away from zero. That would be values greater than 2 or less than −2 . −5 0 5 0 5 0 5 Example: Graph the solution for x ≤ 3 . −5 Example: Graph the solution for x ≥ 3 . −5 McDougal Littell, Chapter 3 HS Pre-Algebra , Unit 03: Multi-Step Equations and Inequalities Revised 2013 - CCSS Page 14 of 14