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www.njctl.org
2015-01-07
6th Grade Math
Equations & Inequalities
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Table of Contents
Inverse Operations
Solving One Step Addition & Subtraction Equations
Click on a topic to go to that section.
Determining Solutions to Equations
Writing Simple Inequalities
Common Core: 6.EE.5,7,8
Solutions to Simple Inequalities
Graphing Solution Sets to Simple Inequalities
Solving One Step Multiplication & Division Equations
Glossary
Writing Equations
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Sometimes when you subtract the fractions, you find that you can't because the first numerator is smaller than the second! When this happens, you need to regroup from the whole number.
How many thirds are in 1 whole?
How many fifths are in 1 whole?
How many ninths are in 1 whole?
Vocabulary words are identified with a dotted underline.
The underline is linked to the glossary at the end of the Notebook. It can also be printed for a word wall.
(Click on the dotted underline.)
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Back to
Instruction
FactorA whole number that can divide into another number with no remainder.
15 3 5
3 is a factor of 153 x 5 = 15
3 and 5 are factors of 15
1635 .1R
3 is not a factor of 16
A whole number that multiplies with another number to make a third number.
The charts have 4 parts.
Vocab Word1
Its meaning 2
Examples/ Counterexamples
3Link to return to the instructional page.
4
(As it is used in the
lesson.)
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An equation can be compared to a balanced scale.
Both sides need to contain the same quantity in order for it to be "balanced". expression 1 expression 2
= sign
What is an equation?An equation is a mathematical statement containing an equal sign to show that two expressions are equal.
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For example,
9 + 11 + 4 = 6 + 14 + 11 is an equation,
because both sides are equal.
9 + 11 + 4 = 6 + 7 + 11 24 = 24
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We can turn this into an algebraic equation, by substituting any of the numbers with a variable.
Examples:
9 + 11 + x = 6 + 7 + 11 x = 4
9 + 11 + 4 = y + 7 + 11 y = 6
9 + c + 4 = 6 + 7 + c c = 11
9 + 11 + 4 = 6 + 7 + 11
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Use the model to represent the following equation.
Let = 1
Let = xx = 5
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What happens if we add one to the right side of the equation to show
Can you write an equation to represent this model? Explain.
x = 6 ?
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A
B
C
DLet = x
Let = 1
2x = 10
2 = 10x
2 = 8x
2x = 1
1 Which equation does the model represent?
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A
B
C
DLet = 2x
Let = 5
2x + 5 = x + 10
4x + 5 = 2x + 5(10)
4x + 5(5) = x + 10
2(2x) + 25 = 2x + 50
2 Which equation does the model represent?
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3 Which equation does this model represent?
A
B
C
D
2x = 6x
x + 2 = 6x
2 + x = 6
2x = 6
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4 Which equation does this model represent?
A
B
C
D
3x = 2x + 7
x + 3 = 7x
3x + 5 = 2x + 35
15 + x = 10 + 6
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Now create your own equation to model using the scale.Remember, in this model, 1 = 5 .
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You already know how to write expressions to represent situations.
d - 5
Lets review the words that indicate addition, subtraction,
multiplication and division.
For example, Jan had d dollars, and she spent 5 dollars.
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As a rule of thumb, if you see the words "than" or "from" it means you have to reverse the order
of the two items on either side of the word.
Translate the following expressions. · 8 less than b means b - 8· 3 more than x means x + 3· x less than 2 means 2 - x
Slide to reveal
Remember!
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What is the best way to represent "three times a"?
(3)(a) a3 3 a 3a
When a variable is being multiplied by a number, the number (coefficient) is always written in front of the variable.
Representing Multiplication
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Representing Division
What is the best way to represent "b divided by 12"?
b ÷ 12
b ∕ 12
b12
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You will now use your knowledge to write equations to represent situations.
Writing equations is basically the same thing as writing expressions. The only difference is that there is an equals sign, and that there are two expressions instead of one.
=expression 1 expression 2equals
sign
3y 12
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Since you already know how to translate words into expressions, lets go over words that can be translated
into an equals sign.
Think of situations where you would use an equals sign.
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PULL
List words that indicate
equals
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Underline the word(s) that mean "equals". Then, find the equation
that represents the words.
EquationWordsfour times a number is 12
12 is 4 less than a number
a number divided by 12 gives you 4
12 is the same value as a number plus 4
n/12 = 412 = n - 4 4n = 12 12 = n + 4
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5 Which equation represents seven minus five is six less than a number.
A
B
C
D
7 - 5 = n - 6
7 - 5 = 6 - n
75 = 6 - n
7 = 5 + n -6
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6 Which equation represents six less than a number comes to the sum of three and seven.
A
B
C
D
n - 6 = 3 + 7
6 - n = 3 + 7
7 - 3 = n + 6
3 + 6 = n - 7
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7 Which equation represents ten times a number totals sixty plus twenty.
A
B
C
D 10n = 60 + 20
n10 = 6 + 20
10n + 60 = 20
60 + 20 + 10 = n
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8 Which equation represents twenty plus four is the same as the product of fourteen and a number.
A
B
C
D
14n = 2 + 4
20 + 4 = 14n
24 = 14n
20 + 4 x 14 = n
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You will now use your knowledge of writing equations to write an equation for real-life scenarios.
George is buying video games online. The cost of the video is $30.00 per game. He spent a total of $127.00. How many games did he buy in all?
Lets pull out the pieces of information, and put them in place.
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George is buying video games online. The cost of the video is $30.00 per game. He spent a total of $127.00. How many games did he buy in all?
Notice that the video games are "per game". We are never told how many games he bought. So we use a variable to represent the number of games. Lets use "g".
· $30.00 per game translates to 30g
· He spent a total of $127.00 translates to = 127
· How many games did he buy in all? means that we are solving for "g".
We know that total means equals.
This is the question we need to answer.
click
click
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30cost of
one videogame number
of games
= 127
totalsamount he spent
g
Lets put it all together
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9 Alice has the 5 newest DVDs, which is 4 less than the amount Jon has. Which equation represents the number of DVDs Jon has.
A
B
C
D
5 = n - 4
5 - 4 = n
n + 5 = 4
4 - n = 9
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10 Mike has $12, which is half as much as Paul has. Which equation represents the amount of money that Paul has?
A
B
C
D 12 = 1 2 p
2p = 12
12 (2) = p
12/p = 2
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11 Jasmine, who bought $5 worth of candy, spent $3 more than Leah spent. Which equation represents the amount that Leah spent?
A
B
C
D
5 = x + 3
x - 3 = 5
x 3 = 5
5 + x = 3
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12 Kate's 93 on her quiz retake was 14 points higher than her original grade. Does the equation 93 = x - 14 correctly represent this scenario?
Yes
No
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13 James is 3 times as old as Thomas, who is 8 years old. Does the equation j = 3(8) correctly represent James' age?
Yes
No
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14 Two brothers put their money together to buy a $19 video game. One contributed $8. Does the equation 8 + x = 19 correctly represent the amount the other brother contributed?
Yes
No
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15 Two brothers split the cost of a $24 video game. Does the equation 2x = 24 correctly represent the amount that each brother contributed?
Yes
No
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What is an equation?
An equation is a mathematical statement, in symbols, that two expressions are exactly the same (or equivalent).
Equations are written with an equal sign, as in:
expression 1 2 + 3 = 5 expression 2
expression 1 9 - 2 = 7 expression 2
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Equations can also be used to state the equality of two expressions containing one or more variables.
In real numbers we can say, for example, that for any given value of x it is true that:
4x + 1 = 14 - 1
If x = 3, then
4(3) + 1 = 14 - 1 12 + 1 = 13 13 = 13
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An equation can be compared to a balanced scale.
Both sides need to contain the same quantity in order for the scale to be "balanced."
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For example, 20 + 30 = 50 represents an equation because both sides simplify to 50.
20 + 30 = 50 50 = 50
Any of the numerical values in the equation can be represented by a variable.
Examples:
20 + c = 50
x + 30 = 50
20 + 30 = y
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A solution to an equation is a number that makes the equation true.
In order to determine if a number is a solution, replace the variable with the number and evaluate the equation.
If the number makes the equation true, it is a solution.If the number makes the equation false, it is not a solution.
Determining the Solutions of Equations
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Example:
Which of the following is a solution of the equation?
y + 12 = 31 {17, 18, 19, 20}
Write the equation four times. Each time replace y with one of the possible solutions and simplify to see if it is true.
17 + 12 = 31 18 + 12 = 31 19 + 12 = 31 20 + 12 = 31 29 = 31 30 = 31 31 = 31 32 = 31 No No Yes No
Answer:19 is the solution to y + 12 = 31
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Try This:
Which of the following is a solution of the equation?
2x + 4 = 18 {4, 5, 6, 7}
Write the equation four times. Each time replace x with one of the possible solutions and simplify to see if it is true.
2(4) + 4 = 18 2(5) + 4 = 18 2(6) + 4 = 18 2(7) + 4 = 18 8 + 4 = 18 10 + 4 = 18 12 + 4 = 18 14 + 4 = 18 12 = 18 14 = 18 16 = 18 18 = 18 No No No Yes
Answer:7 is the solution to 2x + 4 = 18
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Try This:
Which of the following is a solution of the equation?
3y - 4 = 29 {10, 11, 12, 13}
Write the equation four times. Each time replace x with one of the possible solutions and simplify to see if it is true.
3(10) - 4 = 29 3 (11) - 4 = 29 3 (12) - 4 = 29 3 (13) - 4 = 29 30 - 4 = 29 33 - 4 = 29 36- 4 = 29 39 - 4 = 29 26 = 29 29 = 29 32 = 29 35 = 29 No Yes No No
Answer:11 is the solution to 3y - 4 = 29
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Why are we moving on to Solving Equations?
First we evaluated expressions, where we were given the value of the variable, and determined which solution made the equation true.
Now, we are told what the expression equals and we need to find the value of the variable.
When solving equations, the goal is to isolate the variable on one side of the equation in order to determine its value (the value that makes the equation true).
This will eliminate the guess & check of testing possible solutions.
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In order to solve an equation containing a variable, you need to use inverse operations.
Inverse operations are operations that are opposites, or undo one another.
Can you name the inverse of each operation?
Addition - Subtraction
Subtraction - Addition
Multiplication - Division
Division - Multiplication
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When solving equations we are going to use the 4 basic inverse operations:
Addition Subtraction
Multiplication Division
Can you think of any others?
Squaring Square Root
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There are four properties of equality (Addition, Subtraction, Multiplication & Division) that we will use to solve equations.
In simple terms, the properties of equality state when you perform an operation on one side of an equation, you must do the same on the other side of the equation to make sure it stays balanced.
In other words, you can add/subtract/multiply/divide both sides of an equation by the same number and it remains a balancedwithout changing the solution of the equation.
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For each equation, write the inverse operation needed to solve for the variable.
a.) y + 7 = 14 subtract 7 b.) a - 21 = 10 add 21
c.) 5s = 25 divide by 5 d.) x = 5 multiply by 12 12
tap tap
tap tap
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Solving One StepAddition & Subtraction
Equations
Return to Table of Contents
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To solve equations, you must use inverse operations in order to isolate the variable on one side of the equation.
Whatever you do to one side of an equation, you MUST do to the other side!
+5+5
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Examples:
y + 9 = 16 - 9 -9 The inverse of adding 9 is subtracting 9 y = 7
m - 16 = 4 +16 +16 The inverse of subtracting 16 is adding 16 m = 20
Remember - whatever you do to one side of an equation, you MUST do to the other!!!
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x + 8 = 12 - 8 - 8 x = 4
x + 2 = 14 -2 -2 x = 12
x + 5 = 13 - 5 - 5 x = 8
One Step EquationsSolve each equation then click the box to see work & solution.
x - 23 = 43 +23 +23 x = 66
x - 18 = 51 +18 +18 x = 69
x - 4 = 7 +4 +4 x = 11
click to showinverse operation
click to showinverse operation
click to showinverse operation
click to showinverse operation
click to showinverse operation
click to showinverse operation
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Solving One Step Multiplication & Division
EquationsReturn to Table of Contents
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Examples:
6m = 72 6 6 The inverse of multiplying by 6 is dividing by 6 m = 12
2 x m = 3 x 2 The inverse of dividing by 2 is multiplying by 2 2 m = 6
Remember - whatever you do to one side of an equation, you MUST do to the other!!!
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3x = 15 3 3 x = 5
4x = 12 4 4 x = 3
25 = 5x 5 5 5 = x
x = 12 2 2x = 12 x 2 2 x = 24
x = 75 5x = 7 x 55 x = 35
4 = x 66 x 4 = 6x 6 24 = x
One Step EquationsSolve each equation then click the box to see work & solution.
click to showinverse operation
click to showinverse operation
click to showinverse operation
click to showinverse operation
click to showinverse operation
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What do these symbols mean?
LessThan
Less Than or Equal To
Greater Than
Greater Than or Equal To
move square to reveal answer
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An inequality is a statement that two quantities are not equal. The quantities are compared by using one of the following signs:
Symbol Expression Words
< A < B A is less than B
> A > B A is greater than B
< A < B A is less than orequal to B
> A > B A is greater than orequal to B
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When am I ever going to use it?
Your parents and grandparents want you to start eating a healthy breakfast. The table shows the nutritional requirements for a healthy breakfast cereal with milk.
Healthy Breakfast Cereals (per serving)
Fat Less than 3 grams
Protein More than 5 grams
Fiber At least 3 grams
Sugar At most 5 grams
Suppose your favorite cereal has 2 grams of fat, 7 grams of protein, 3 grams of fiber and 4 grams of sugar. Is it a healthy cereal?
Ans
wer
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Is a cereal with 3 grams of fiber considered healthy?
Ans
wer
Healthy Breakfast Cereals (per serving)
Fat Less than 3 grams
Protein More than 5 grams
Fiber At least 3 grams
Sugar At most 5 grams
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Is a cereal with 5 grams of sugar considered healthy?
Ans
wer
Healthy Breakfast Cereals (per serving)
Fat Less than 3 grams
Protein More than 5 grams
Fiber At least 3 grams
Sugar At most 5 grams
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When you need to use an inequality to solve a word problem, you may encounter one of the phrases below.
Important Words
Sample Sentence Equivalent Translation
is more than Trenton is more than 10 miles away.
d > 10
is greater than A is greater than B. A > B
must exceed The speed must exceed 25 mph.
The speed is greater than 25 mph.
s > 25
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Here are some more expressions you may encounter:
Important Words
Sample Sentence Equivalent Translation
cannot exceedTime cannot exceed 60 minutes.
Time must be less than or equal to 60 minutes.
t < 60
is at mostAt most, 7 students were late for class.
Seven or fewer students were late for class.
n < 7
is at least Bob is at least 14 years old.
Bob's age is greater than or equal to 14.
B > 14
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How are these inequalities read?
2 + 2 > 3 Two plus two is greater than 3
2 + 2 ≥ 4 Two plus two is greater than or equal to 4
2 + 2 < 5 Two plus two is less than 5
2 + 2 ≤ 5 Two plus two is less than or equal to 5
2 + 2 ≤ 4 Two plus two is less than or equal to 4
2 + 2 > 3 Two plus two is greater than or equal to 3
Click to Reveal
Click to Reveal
Click to Reveal
Click to Reveal
Click to Reveal
Click to Reveal
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Writing inequalitiesLet's translate each statement into an inequality.
x is less than 10
20 is greater than or equal to y
x < 10
words
inequality statement
translate to
20 > y
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Try These:
1. 14 is greater than a
2. b is less than or equal to 8
3. 6 is less than the product of f and 20
4. The sum of t and 9 is greater than or equal to 36
5. 7 more than w is less than or equal to 10
6. 19 decreased by p is greater than or equal to 2
7. Fewer than 12 items
8. No more than 50 students
9. At least 275 people attended the play
Ans
wer
s
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Try to change the following expressions from English into math.
Twice a number is at most six.
Two plus a number is at least four.
2x ≤ 6
2 + x ≥ 4
Answer
Answer
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Three less than a number is less than five.
The product of a number and thirteen is greater than nine.
Three times a number plus one is at least ten.
x - 3 < 5
13x > 9
3x + 1 > 10
Answer
Answer
Answer
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Remember: Equations have one solution.
Solutions to inequalities are NOT single numbers. Instead, inequalities will have more than one value for a solution.
10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10
This would be read as, "The solution set is all numbers greater than or equal to negative 5."
Solution Sets
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Let's name the numbers that are solutions to the given inequality.
r > 10 Which of the following are solutions? {5, 10, 15, 20}
5 > 10 is not trueSo, 5 is not a solution
10 > 10 is not trueSo, 10 is not a solution
15 > 10 is trueSo, 15 is a solution
20 > 10 is trueSo, 20 is a solution
Answer:{15, 20} are solutions to the inequality r > 10
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Let's try another one.
30 ≥ 5d; {4,5,6,7,8}
30 ≥ 5d30 ≥ 5(4)30 ≥ 20
30 ≥ 5d30 ≥ 5(5)30 ≥ 25
30 ≥ 5d30 ≥ 5(6)30 ≥ 30
30 ≥ 5d30 ≥ 5(7)30 ≥ 35
30 ≥ 5d30 ≥ 5(8)30 ≥ 40
Answer: {4,5,6}
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Since inequalities have more than one solution, we show the solution two ways.
The first is to write the inequality. The second is to graph the inequality on a number line.
In order to graph an inequality, you need to do two things:
1. Draw a circle (open or closed) on the number that is your boundary.
2. Extend the line in the proper direction.
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Determining Whether to Use an Open or Closed Circle
An open circle on a number shows that the number is not part of the solution. It serves as a boundary only.
It is used with "greater than" and "less than".The word equal is not included.< >
A closed circle on a number shows that the number is part of the solution.
It is used with "greater than or equal to" and "less than or equal to".< >
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Determining Which Direction to Extend the Line
Extend Line to the Left: If the variable is smaller than the number then you extend your line to the left (since smaller numbers are on the left).
Extend the line to the left in these situations:# > variablevariable < #
Extend Line to the Right: If the variable is larger than the number then you extend your line to the right (since bigger numbers are on the right).
Extend the line to the right in these situations:# < variablevariable > #
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Graph the solution to: x is less than one
Step 1: Figure out what the inequality solution requires. For example, rewrite x is less than one as x < 1.
Step 2: Draw a circle on the number line where the number being graphed is represented. In this case, draw an open circle since it represents the starting point for the inequality solution but is not part of the solution.
-1 0-2-3-4-5 1 2 3 4 5
Graphing Inequalities
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Step 4: Draw a line, thicker than the horizontal line, from the dot to the arrow. This represents all of the numbers that fulfill the inequality.
Step 3: Draw an arrow on the number line showing all possible solutions. This number is less than one, so the arrow will be drawn to the left of the boundary point.
-1 0-2-3-4-5 1 2 3 4 5
-1 0-2-3-4-5 1 2 3 4 5
x < 1
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10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10
Example
Graph the solution to: x is greater than or equal to one
Step 1: Rewrite x is greater than or equal to one as x > 1.
Step 2: Draw a circle on the number line at 1. In this case, a closed circle since it represents the starting point and is a part of the solution.
Step 3: Determine which direction to draw your arrow and extend your line. Since x is greater than 1, you will extend your line to the right.
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Remember!
Open circle means that number is not included in the solution set and is used to represent < or >.
Closed circle means the solution set includes that number and is used to represent ≤ or ≥.
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10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10
Try These.
Graph the inequality.x > 5
Graph the inequality -3 > x
10 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8-9-10
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Try these.Graph the inequalities.
1. x > 4
-1 0-2-3-4-5 1 2 3 4 5
2. x < -5
-1 0-2-3-4-5 1 2 3 4 5
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Try these.State the inequality shown.
1. x < 5
-1 0-2-3-4-5 1 2 3 4 5
-1 0-2-3-4-5 1 2 3 4 5
2. x > -1
Click to Reveal
Click to Reveal
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Remember!
Closed circle means the solution set includes that number and is used to represent ≤ or ≥.
Open circle means that number is not included in the solution set and is used to represent < or >.
Extend your line to the right when the variable is larger than the number. # < variable variable > #
Extend your line to the left when the variable is smaller than the number. # > variable variable < #
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0 1 2 3 4 5 6 7 8 9 10
7.5
$7.50
7.5
at least
>
An employee earns
e
A store's employees earn at least $7.50 per hour. Define a variable and write an inequality for the amount the employees may earn per hour. Graph the solutions.
Let e represent an employee's wages.
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Try this:
The speed limit on a road is 55 miles per hour. Define a variable, write an inequality and graph the solution.
Ans
wer
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Back to
Instruction
EquationTwo expressions that are equivalent to each other. Equivalence is shown
with an equal sign.
4x= 8equivalent
expressions4 =x3
equivalent expressions
no equivalence
x3
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Back to
Instruction
ExpressionNumbers, symbols and
operations grouped together that show the value of something.
2 x 3 = 6 Expressions DO NOT have
an equal signs.
3 2 + 12 An expression is one side of an equation.
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Back to
Instruction
InequalityA comparison of two numbers that
are not, or may not, be equal.
larg
er
smaller
larg
ersmallerGreater
thanLess than
Greater than or equal to
Less than or equal
to
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Back to
Instruction
Inverse Operations
+-
x+
-
Addition reverses
subtraction
Subtraction reverses addition
x
Division reverses
multiplication
Multiplication reverses division
x
x
Squaring reverses
square roots
Square roots reverses squaring
932
932
Operations that "undo" each other. They are opposites.
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Back to
Instruction
Isolate the Variable
To move the variable to one side of the equation, and all of the
numbers to the other side.To isolate
the variable, use inverse operations
& properties of numbers.
Expression 1: Only the variable2y = 12
y = 6Expression 2: Everything else
Inverse operation
x
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Back to
Instruction
SolutionA value you can put in place of a
variable that would make the statement true.
x + 4 = 9Solution:
x = 5
The answer to
a math problem.
3y 6 Solution:
y 2
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Back to
Instruction
Solution Set
A set of values that can make a statement true.
The #s in a solution set are written in curly
brackets.
{ }y2 = 16y = {4,-4}
3 < y < 7{4,5,6,7}y=
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Back to
Instruction
VariableA letter or symbol that represents a
changeable or unknown value.
4x + 2
variablex = ?
2x = 6x x
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Back to
Instruction
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Instruction
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Back to
Instruction
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Back to
Instruction
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