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 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 How to Identify Multistep Equations Characteristic Example Multiple variable or constant terms on the same side x + 2x + 3x – 1 = Variable present on both sides 4x – 2 = 3x + 3 Parentheses present on either side 3(x + 2) = 21

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 –Combine like terms Ex) 5x – 10 + x = 2x –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 – = 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 Combining Terms

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

Same general set of steps is useful in solving many multistep equations How to Solve Multistep Equations Steps to Solve a Multistep Equation First step Simplify each side as much as possible 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? Justify They each began with $35 Evaluate 35 – 5 is double 35 – 20 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

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

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? Justify Evaluate 14, 15, and 16 are consecutive integers and add to 45 Analyze Formulate Represent the problem as an equation Determine n + (n + 1) + (n + 2) = 45 n + n n + 2 = 45 3n + 3 = 45 3n = 42 n = 14, n + 1 = 15, n + 2 = 16

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

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? Justify Evaluate Reasonable as the sum of the four integers 17, 19, 21, and 23 adds to 80 Analyze 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 n n + 6 = 80 4n + 12 = 80 4n = 68 n = 17, so n + 6 = 23

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

Multistep Inequalities Example Ex) Find the solution set of the inequality 3(x + 2) > 5x. Justify Evaluate x < 3 describes a range of values that satisfies an inequality Analyze Formulate 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

Multistep Inequalities Example Ex) Find the solution set of the inequality 4x – 6 < 2(x + 1). Graph the solution on a number line. Justify Evaluate Reasonable because it describes a range of values that satisfy the inequality Analyze Asks for graph of solution Formulate Determine 4x – 6 < 2(x + 1) 4x – 6 < 2x + 2 2x – 6 < 2 2x < 8 x < 4

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 Multistep Equations