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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