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Multistep Equations How to Identify Multistep Equations |Combining Terms| How to Solve Multistep Equations | Consecutive Integers | Multistep Inequalities Multistep Equations Learning Objectives • Use the properties of equality to solve multistep equations of one unknown • Apply the process of solving multistep equations to solve multistep inequalities How to Identify Multistep Equations • Some equations can be solved in one or two steps – Ex) 4x + 2 = 10 is a two-step equation • Subtract 2 • Divide both sides by 4 – Ex) 2x – 5 = 15 is a two-step equation • Add 5 • Divide both sides by 2 How to Identify Multistep Equations • Multistep equation – an equation whose solution requires more than two steps – Ex) 5x – 4 = 3x + 2 and 4(x – 2) = 12 • Multistep equations can take different forms – Variables present in two different terms • Ex) 6x – 2x = 8 + 4, 5x – 4 = 3x + 2 • Ex) 4(x – 2) = 12 Characteristic Example Multiple variable or constant terms on the same side Variable present on both sides x + 2x + 3x – 1 = 3 + 20 Parentheses present on either side 3(x + 2) = 21 4x – 2 = 3x + 3 Combining Terms • First step to solving a multistep equation is to simplify each side – Use the distributive property to eliminate parentheses • Ex) 5(x – 2) + x = 2(x + 3) + 4 simplifies to 5x – 10 + x = 2x + 6 + 4 – Combine like terms • Ex) 5x – 10 + x = 2x + 6 + 4 – Combine 5x and x to 6x on the left side – Combine 6 and 4 to 10 on the right side – Simplifies to 6x – 10 = 2x + 10 • Terms containing variables cannot be combined with constant terms Combining Terms • Consider 5(x – 3) + 4 = 3(x + 1) – 2 – Distributive Property can be used on both sides due to the parentheses multiplied by constants • Produces 5x – 15 + 4 = 3x + 3 – 2 – Terms can be combined on both sides • Combine –15 and 4 on the left and 3 and –2 on the right • Produces 5x – 11 = 3x + 1 How to Solve Multistep Equations • Consider the equation 4(x + 2) – 10 = 2(x + 4) – Simplify with the distributive property • 4x + 8 – 10 = 2x + 8 – Combine like terms • 4x – 2 = 2x + 8 – Eliminate the unknown from one side • 2x – 2 = 8 – Eliminate the constant term on the other side • 2x = 10 – Divide each side by the coefficient of the variable • x=5 How to Solve Multistep Equations Steps to Solve a Multistep Equation Simplify each side First step as much as possible • Same general set of steps is useful in solving many multistep equations Second step Eliminate the variable from one side Third step Eliminate the constant term from the side with the variable Fourth step Divide each side by the coefficient of the variable How to Solve Multistep Equations Example Ex) Mark and Susan are given the same amount of money. Mark spends $5, and Susan spends $20. If Mark now has twice as much money as Susan, how many dollars did they each have originally? Analyze Find the original amounts Formulate Represent the problem as an equation Determine x – 5 = 2(x – 20) x – 5 = 2x – 40 –x – 5 = –40 –x = –35 x = 35 Justify They each began with $35 Evaluate 35 – 5 is double 35 – 20 Consecutive Integers • Consecutive integers – integers that are separated by exactly one unit – Ex) 5 and 6 • Set of consecutive integers from –1 to 3 • When domain is limited to integers, the letter n is used to represent the variable – Ex) Find 3 integers that sum to 24 • Solution is 7, 8, 9 Consecutive Integers • Ex) Find three consecutive integers where the sum of the first two is equal to 3 times the third one – Can be represented as n + (n + 1) = 3(n + 2) – Can be solved using the process to solve multistep equations Consecutive Integers Example Ex) Mrs. Smith has three children whose ages are spaced one year apart. How old will they be when their ages add to 45? Analyze Justify Formulate Evaluate Represent the problem as an equation Determine n + (n + 1) + (n + 2) = 45 n + n + 1 + n + 2 = 45 3n + 3 = 45 3n = 42 n = 14, n + 1 = 15, n + 2 = 16 14, 15, and 16 are consecutive integers and add to 45 Consecutive Integers • Consecutive even integers are spaced two units apart – Ex) 6, 8, and 10 – Can be represented as n, n + 2, n + 4 • Consecutive odd integers represented the same way – Ex) 7, 9, and 11 – Can be represented as n, n + 2, n + 4 • Check answer to make sure the solutions are odd Consecutive Integers Example Ex) Four brothers sit next to each other at a baseball game, and they notice that the seat numbers are all odd in their section. If their seat numbers add to 80, what is the greatest of the seat numbers? Analyze Justify Find the greatest seat number Formulate Represent the problem as an equation and solve it Determine n + (n + 2) + (n + 4) + (n + 6) = 80 n + n + 2 + n + 4 + n + 6 = 80 4n + 12 = 80 4n = 68 n = 17, so n + 6 = 23 Evaluate Reasonable as the sum of the four integers 17, 19, 21, and 23 adds to 80 Multistep Inequalities • Multistep inequalities can be solved in the same way as multistep equations – Ex) 6(x – 2) > 2x can be solved with the four step process for multistep equations • • • • First, simplify each side, 6x – 12 > 2x Second, eliminate variable on right side, 4x – 12 > 0 Third, eliminate constant from left side, 4x > 12 Last, divide each side by coefficient of unknown, x > 3 – Ex) 2(x + 6) ≤ 6x • • • • First, simplify each side, 2x + 12 ≤ 6x Second, eliminate variable on right side, –4x + 12 ≤ 0 Third, eliminate constant from left side, –4x ≤ –12 Last, divide each side by coefficient of unknown, x ≥ 3 Multistep Inequalities • Sometimes it is simpler to put the variable term on the right side instead of the left – Ex) 2x + 12 ≤ 6x can be made into a two-step inequality • Subtract 2x from each side to produce 12 ≤ 4x • Divide by 4 to yield 3 ≤ x • Placing the variable on the left also requires that the inequality sign be flipped, resulting in x ≥ 3 • Numbers in the solution range, such as 5, can illustrate that the statements 3 ≤ x and –x ≤ –3 are equivalent Multistep Inequalities Example Ex) Find the solution set of the inequality 3(x + 2) > 5x. Analyze Justify Formulate Evaluate Combine unknowns on the left side Determine 3(x + 2) > 5x 3x + 6 > 5x 3x – 5x + 6 > 0 –2x + 6 > 0 –2x > – 6 x<3 x < 3 describes a range of values that satisfies an inequality Multistep Inequalities Example Ex) Find the solution set of the inequality 4x – 6 < 2(x + 1). Graph the solution on a number line. Analyze Justify Asks for graph of solution Formulate Determine 4x – 6 < 2(x + 1) 4x – 6 < 2x + 2 2x – 6 < 2 2x < 8 x<4 Evaluate Reasonable because it describes a range of values that satisfy the inequality Multistep Equations Learning Objectives • Use the properties of equality to solve multistep equations of one unknown • Apply the process of solving multistep equations to solve multistep inequalities