How to Identify Multistep Equations |Combining
Terms| How to Solve Multistep Equations |
Consecutive Integers | Multistep Inequalities

• Some equations can be solved in one or two steps
– Ex) 4 x + 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

• Multistep equation – an equation whose solution requires more than two steps
– Ex) 5 x – 4 = 3 x + 2 and 4( x – 2) = 12
• Multistep equations can take different forms
– Variables present in two different terms
• Ex) 6 x – 2 x = 8 + 4, 5 x – 4 = 3 x + 2
• Ex) 4( x – 2) = 12
Characteristic
Multiple variable or constant terms on the same side
Variable present on both sides
Parentheses present on either side
Example x + 2 x + 3 x – 1 = 3 + 20
4 x – 2 = 3 x + 3
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 5 x – 10 + x = 2 x + 6 +
4
– Combine like terms
• Ex) 5 x – 10 + x = 2 x + 6 + 4
– Combine 5 x and x to 6 x on the left side
– Combine 6 and 4 to 10 on the right side
– Simplifies to 6 x – 10 = 2 x + 10
• Terms containing variables cannot be combined with constant 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 5 x – 15 + 4 = 3 x + 3 – 2
– Terms can be combined on both sides
• Combine –15 and 4 on the left and 3 and –2 on the right
• Produces 5 x – 11 = 3 x + 1

• Consider the equation 4( x + 2) – 10 = 2( x + 4)
– Simplify with the distributive property
• 4 x + 8 – 10 = 2 x + 8
– Combine like terms
• 4 x – 2 = 2 x + 8
– Eliminate the unknown from one side
• 2 x – 2 = 8
– Eliminate the constant term on the other side
• 2 x = 10
– Divide each side by the coefficient of the variable
• x = 5

• Same general set of steps is useful in solving many multistep equations
Steps to Solve a Multistep
Equation
First step
Simplify each side as much as possible
Second step
Third step
Fourth step
Eliminate the variable from one side
Eliminate the constant term from the side with the variable
Divide each side by the coefficient of the variable

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 = 2 x – 40
–x – 5 = –40
– x = –35 x = 35
Justify
They each began with $35
Evaluate
35 – 5 is double 35 – 20
End Part 1

• 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

• 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

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

• 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

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

• Multistep inequalities can be solved in the same way as multistep equations
– Ex) 6( x – 2) > 2 x can be solved with the four step process for multistep equations
• First, simplify each side, 6 x – 12 > 2 x
• Second, eliminate variable on right side, 4 x – 12 > 0
• Third, eliminate constant from left side, 4 x > 12
• Last, divide each side by coefficient of unknown, x > 3
– Ex) 2( x + 6) ≤ 6 x
• First, simplify each side, 2 x + 12 ≤ 6 x
• Second, eliminate variable on right side, –4 x + 12 ≤ 0
• Third, eliminate constant from left side, –4 x ≤ –12
• Last, divide each side by coefficient of unknown, x ≥ 3

• Sometimes it is simpler to put the variable term on the right side instead of the left
– Ex) 2 x + 12 ≤ 6 x can be made into a two-step inequality
• Subtract 2 x from each side to produce 12 ≤ 4 x
• 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

Ex) Find the solution set of the inequality 3(x + 2) > 5x.
Analyze Justify
Formulate
Combine unknowns on the left side
Determine
3( x + 2) > 5 x
3 x + 6 > 5 x
3 x – 5 x + 6 > 0
–2 x + 6 > 0
–2 x > – 6 x < 3
Evaluate x < 3 describes a range of values that satisfies an inequality

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