1. Notes
7th Grade Math
McDowell
Chapter 2

2. ONE STEP EQUATIONS 9/25 equations
An equation is like a balance scale because it shows that two quantities are equal.
What you do to one side of the equation must also be done to the other side to keep it balanced.

3. Addition property of equality
Add the same thing on both sides to keep the equation equal.
Subtraction property of equality
Subtract the same thing on both sides to keep the equation equal.

4. Ask three questions about the equation:
solving one step equations
Ask three questions about the equation:
1. What is the variable?
2. What operation is performed on the variable?
3. What is the inverse operation?
(The one that will undo what is being done to the variable)

5. Solve x + 4 = 12 Example x Addition. Subtraction. x + 4 = 12 - 4 - 4 x
1. What is the variable?
2. What operation is being performed on the variable?
Addition.
3. What is the inverse operation (the one that will undo what is being done to the variable)?
Subtraction.
The subtraction property of equality tells us to subtract the same thing on both sides to keep the equation equal.
x + 4 = 12
- 4
- 4 x = 8

6. Solve y - 7 = -13 Example y Subtraction. Addition. y - 7 = -13 + 7 + 7
1. What is the variable?
2. What operation is being performed on the variable?
Subtraction.
3. What is the inverse operation (the one that will undo what is being done to the variable)?
Addition.
The addition property of equality tells us to add the same thing on both sides to keep the equation equal.
y - 7 = -13
+ 7
+ 7 y = -6

7. You Try
Workbook
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8. Multiply the same thing on both sides to keep the equation equal.
9/28
Multiplication property of equality
Multiply the same thing on both sides to keep the equation equal.
Division property of equality
Divide the same thing on both sides to keep the equation equal.

9. Solve –6a = 12 Example a Multiplication. Division –6a = 12 -6 -6 a =
1. What is the variable?
2. What operation is being performed on the variable?
Multiplication.
3. What is the inverse operation (the one that will undo what is being done to the variable)?
Division
The division property of equality tells us to divide the same thing on both sides to keep the equation equal.
–6a = 12 -6 -6 a = -2

10. -10 Solve = b Division. Multiplication = -10 2 • • 2 b = -20
1. What is the variable?
2. What operation is being performed on the variable?
Division.
3. What is the inverse operation (the one that will undo what is being done to the variable)?
Multiplication
The multiplication property of equality tells us to multiply the same thing on both sides to keep the equation equal.
= -10
2 •
= -10
• 2 b =
-20

11. You Try
Workbook
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12. Evaluating and Writing Algebraic Expressions 9/29
Variable
Letters that stand for numbers
x + 4 = 5, x is 1
Variables are used when we don’t know the number

13. Expression
A mathematical phrase
No = sign
3 + 7
Algebraic
Expression
A mathematical phrase that uses variables, numerals, and operation symbols
4a + b
a and b are both variables

14. Words that mean math Addition Subtraction
increased by more than combined, together total of sum added to
Subtraction
decreased by minus, less difference between/of less than, fewer than

15. Words that mean math Multiplication Division Of, times,
multiplied by, product of, increased/decreased
by a factor of
Division
per, a, out of,
ratio of, quotient of,
percent (divide by 100)

16. Words that mean math Equals
is, are, was, were, will be gives, yields sold for

17. Examples 12 inches per foot 12 in / ft 2. 10 packs of x crayons 10x
3. 7 less than t
t – 7
(this one is written backwards from the word order—be careful)

18. Reminder Addition and multiplication are commutative
Order is not a issue
Subtraction and division are not
Order matters A LOT!
Reminder

19. You Try The value in cents of q quarters
13° warmer than the room temperature t
The sum of 18 and b
5 less than a
The quotient of d and 6

20. Evaluating Expressions 9/29
Evaluate an expression
Simplify the expression by combining like terms
Combine numbers with numbers and like variables with like variables
Evaluate a variable expression
1. Plug in the number the variable is standing in for
2. Simplify by combining like terms

21. Plug Chug & Evaluate a variable expression
in the number the variable is standing in for
Plug
&
chug chug out the answer by combining like terms
Chug

22. Example Plug Chug Evaluate 2x + x for x = 4 2(4) + 4 8 + 4 12
In the number
2(4) + 4
Chug
Out the answer
8 + 4 12

23. Example Plug Chug Evaluate 2ab for a = 3, b = 4 2(3)(4) 6(4) 24
In the numbers
2(3)(4)
Chug
Out the answer
6(4) 24

24. You Try
Workbook
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25. Evaluating Two-Step Equations 9/30
Two operations are required to isolate x
Plug and Chug to check your answer

26. Ask five questions about the equation:
solving two step equations
1. What is the variable?
2. What is the first operation being done to the variable?
3. What is the inverse operation?
4. What is the second operation being done to the variable?
5. What is the inverse operation?

27. Work Backwards
First—undo any addition or subtraction
Second—undo any multiplication or division

28. Example 2 · · 2 = - 4 x = -8 -4 – 3 = -7 -7 = -7 + 3 + 3
Draw “the road”
Work Backwards
Add 3 to both sides x
 2
• 2 -3 +3
Simplify
2 · · 2
= - 4
Multiply both sides by 2
Simplify
x = -8
Check your answer
Plug and Chug
-4 – 3 = -7
-7 = -7

29. Solve Remember to work backwards!!!! Additional Examples
Addition or subtraction 1st!!
Additional
Examples
Solve
1. 2x - 1 = -3 2.
3. 5z + 16 = 51

30. Video
Interactive

31. You try
Workbook
p 32
# 1-29 odd

32. Graphing and Writing Inequalities 10/1
Inequality signs
<
>
Open dot
Closed dot
less than
fewer than
greater than
more than
exceeds
less than or equal to
no more than
at most
greater than or equal to
no less than
at least

33. Solution of An
Inequality
Any value that makes the inequality true
There can be infinity many solutions
Which are solutions of x  -3?
5, 9, -2, or –37
Replace x with the possible solution and determine if it is true.

34. ≤ Has a closed circle and points left
Graphing
Inequalities
Graph c > -2.
> Has an open circle and points right
Graph m ≤ 4.
≤ Has a closed circle and points left

35. Writing Inequalities from the graph
Write an inequality from each graph
Closed circle and points right 
Circle is over the 1
x  1
Open circle and points left >
Circle is over the -4
x > -4.

36. You Try
Workbook
p 36
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37. Solving Inequalities by Adding or Subtracting 10/20
The
Inequality
sign
When adding or subtracting, treat the inequality sign just like you would an = sign
Perform the inverse to isolate the variable and leave the inequality sign alone.

38. Ask three questions about the inequality:
Reminder
Ask three questions about the inequality:
1. What is the variable?
2. What operation is performed on the variable?
3. What is the inverse operation?
(The one that will undo what is being done to the variable)

39. Solve x + 7 > 15 Example x Addition. Subtraction. x + 7 > 12 - 7
1. What is the variable?
2. What operation is being performed on the variable?
Addition.
3. What is the inverse operation (the one that will undo what is being done to the variable)?
Subtraction.
The subtraction property of inequality tells us to subtract the same thing on both sides to keep the equation equal.
x + 7 > 12
- 7
- 7 x
> 5

40. Example y Subtraction. Addition. y – 10  -12 +10 +10 y  -2
Solve y – 10  -12
Example y
1. What is the variable?
2. What operation is being performed on the variable?
Subtraction.
3. What is the inverse operation (the one that will undo what is being done to the variable)?
Addition.
The addition property of inequality tells us to add the same thing on both sides to keep the equation equal.
y – 10  -12
+10
+10 y  -2

41. You Try
Workbook
p 38
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42. Solving Inequalities by Multiplying or Dividing 10/23
When multiplying or dividing on both sides of an inequality sign watch for negatives.
The
Inequality
sign
When multiplying or dividing by a positive number leave the inequality sign alone.
When multiplying or dividing by a negative number, flip the inequality sign.

43. Solve 2a < 10 Example a Multiply by 2 Divide by 2 2a < 10 2 2
1. What is the variable?
2. What operation is being performed on the variable?
Multiply by 2
3. What is the inverse operation (the one that will undo what is being done to the variable)?
Divide by 2
The division property of inequality tells us to divide the same thing on both sides to keep the equation equal.
2a < 10
2 is positive so leave the inequality sign alone. 2 2 a
< 5

44. Solve m/-4  3 Example m Divide by -4 Multiply by -4 m/-4  3 X-4 X-4
1. What is the variable?
2. What operation is being performed on the variable?
Divide by -4
3. What is the inverse operation (the one that will undo what is being done to the variable)?
Multiply by -4
The division property of inequality tells us to divide the same thing on both sides to keep the equation equal.
m/-4  3
-4 is negative so flip the inequality sign.
X-4
X-4 m 
-12

45. You Try
Workbook
p 40
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46. Evaluating Two-Step Inequalities 10/27
Two operations are required to isolate x
Plug and Chug to check your answer
Remember: When multiplying or dividing by a negative number, flip the inequality sign.

47. Ask five questions about the inequality:
Solving Two Step Inequalities
1. What is the variable?
2. What is the first operation being done to the variable?
3. What is the inverse operation?
4. What is the second operation being done to the variable?
5. What is the inverse operation?

48. Work Backwards
First—undo any addition or subtraction
Second—undo any multiplication or division

49. Example -2x – 5  25  30 x  -15 -2(-15) – 5  25 30 – 5  25 25  25
Draw “the road”
Example
-2x – 5  25
Work Backwards
Add 5 to both sides
Simplify
-2x
 30 x
• -2
 -2 -5 +5 -2 -2
Divide both sides by -2
Simplify
x  -15
When multiplying or dividing by a negative number, flip the inequality sign.
-2(-15) – 5  25
30 – 5  25
Check your answer
Plug and Chug
25  25
Graph
-17
-16
-15
-14
-13

50. You Try
Simplify and graph
y – 4 > -12 2
2. –4m + 10  30
3. ¼a – 13 < -3

51. You Try
Workbook
p 40 all