1. Exercise
Simplify.
5 + 7 − 7 5

2. Exercise
Simplify.
−12 −
−12

3. Exercise
Simplify.
136,798 − 3, ,479
136,798

4. Exercise
Simplify.
7 × 8 ÷ 8 7

5. Exercise
Simplify.
375 ÷ 15 × 15
375

6. Equation
An equation is a mathematical sentence that contains an equal sign.

7. Solution
A solution is a number that, when substituted for a variable, makes a mathematical sentence true.

8. Inverse Operations
Inverse operations are operations that undo one another.

9. Addition/Subtraction Multiplication/Division
Inverse Operations
Addition/Subtraction
Multiplication/Division

10. Exercise
Is this an equation? If so, what is the solution?
2x + 3 no

11. Exercise
Is this an equation? If so, what is the solution?
x + 3 = 4
yes; 1

12. Exercise
Is this an equation? If so, what is the solution?
4x2 − 5 no

13. Exercise
Is this an equation? If so, what is the solution?
n − 5 = 5
yes; 10

14. Addition Property of Equality
For all integers a, b, and c, if a = b, then a + c = b + c.

15. Example 1
Solve x − 6 = 10, and check the solution.
x = 16

16. Example 2
Solve a + 39 = 17, and check the solution.
a = -22

17. Solving an Equation Involving Addition or Subtraction
Determine what operation must be performed in order to isolate the variable on one side of the equation.

18. Solving an Equation Involving Addition or Subtraction
Undo addition by subtracting, or undo subtracting by adding.
Remember that what you do to one side of the equation you must do to the other side also.

19. 15 = y + 11
y + 11 = 15

20. Symmetric Property of Equality
If a = b, then b = a.

21. Example 3
Solve 47 = m + 3, and check the solution.
m = 44

22. 3 − (−6)
Subtracting a “−” is the same as

23. 3 + (−6)
Adding a “−” is the same as

24. Example 4
Solve y − (−7) = −18, and check the solution.
y = −25

25. Example 5
Solve z + (−32) = −19, and check the solution.
y = 13

26. Example 6
Write each word phrase as a mathematical expression. Use n for the variable.
a. four more than a number
n + 4
b. a number increased by ten
n + 10

27. Example 6 c. five less than a number n − 5
d. a number decreased by seven
n − 7
e. seventeen decreased by a number
17 − n

28. Example 7
Write an algebraic expression for the number of nickels Sergei has and the number Kimberly has. Sergei has 8 more than Evan, and Kimberly has 6 fewer than Evan. Use x as the variable.

29. Example 7 Let x = the number of nickels Evan has.
Sergei has x + 8 nickels.
Kimberly has x − 6 nickels.

30. Example 8
Write an equation for the sentence “A number decreased by three is sixteen.”
Let n = the number.
n − 3 = 16

31. Example 9
Write an equation for the sentence “Seven dollars more than the cost of the shoes is ninety-eight dollars.”
Let c = the cost.
c + 7 = 98

32. Exercise
Solve.
−5 + (s + 2) = −9(8)

33. Exercise
Solve.
(17 − 12) − (−a) = 125 − 250

34. Exercise
Solve.
−6y y −4 = −8 + (−41)

35. Exercise
Solve.
23 + c − = 7 − 61

36. Exercise
Solve.
d + [17 − (−12)] = 5 − (−58)