1. An equation is a mathematical statement that two expressions are equal.
y=13
X=85

2. Objective
Solve one-step equations in one variable by using multiplication or division.

3. Inverse Operations Operation Inverse Operation
Solving an equation that contains multiplication or division is similar to solving an equation that contains addition or subtraction. Use inverse operations to undo the operations on the variable.
Inverse Operations
Operation
Inverse Operation
Multiplication
Division
Division
Multiplication

4. Example 1A: Solving Equations by Using Multiplication
Solve the equation.
–8 = j 3
Since j is divided by 3, multiply both sides by 3 to undo the division.
–24 = j
Check
–8 = j 3 –8
–24 3
To check your solution, substitute –24 for j in the original equation.
–8 –8 

5. Example 1B: Solving Equations by Using Multiplication
Solve the equation.
= 2.8 n 6
Since n is divided by 6, multiply both sides by 6 to undo the division.
n = 16.8
Check
= 2.8 n 6
2.8
16.8 6
To check your solution, substitute 16.8 for n in the original equation. 

6. Check It Out! Example 1a
Solve the equation. Check your answer.
= 10 p 5
Since p is divided by 5, multiply both sides by 5 to undo the division.
p = 50
Check
= 10 p 5 10 50 5
To check your solution, substitute 50 for p in the original equation. 

7. Example 2A: Solving Equations by Using Division
Solve the equation. Check your answer.
9y = 108
Since y is multiplied by 9, divide both sides by 9 to undo the multiplication.
y = 12
Check 9y = 108
To check your solution, substitute 12 for y in the original equation.
9(12) 108 

8. Check It Out! Example 2a
Solve the equation. Check your answer.
16 = 4c
Since c is multiplied by 4, divide both sides by 4 to undo the multiplication.
4 = c
Check = 4c
16 4(4)
To check your solution, substitute 4 for c in the original equation. 

9. Check It Out! Example 2c
Solve the equation. Check your answer.
15k = 75
Since k is multiplied by 15, divide both sides by 15 to undo the multiplication.
k = 5
Check 15k = 75
To check your solution, substitute 5 for k in the original equation.
15(5) 75 

10. Remember that dividing is the same as multiplying by the reciprocal
Remember that dividing is the same as multiplying by the reciprocal. When solving equations, you will sometimes find it easier to multiply by a reciprocal instead of dividing. This is often true when an equation contains fractions.

11. Example 3A: Solving Equations That Contain Fractions
Solve the equation. 5
w = 20 6
The reciprocal of is Since w is multiplied by , multiply both sides by 5 6
w = 24
Check
w = 20 5 6
To check your solution, substitute 24 for w in the original equation.
20
20 20 

12. Example 3B: Solving Equations That Contain Fractions
Solve the equation. 3 1 8
= z 16
The reciprocal of is 8. Since z is multiplied by , multiply both sides by 8. 1 8
= z 3 2
Check 1 8 3 16
= z
To check your solution, substitute for z in the original equation. 3 2 3 16 

13. Check It Out! Example 3b Solve the equation. 4j 2 = 6 3
is the same as j. 4 6 4j
The reciprocal of is . Since j is multiplied by , multiply both sides by . 4 6
j = 1

14. Write an equation to represent the relationship.
Check it Out! Example 4
The distance in miles from the airport that a plane should begin descending, divided by 3, equals the plane's height above the ground in thousands of feet. A plane began descending 45 miles from the airport. Use the equation to find how high the plane was flying when the descent began.
Distance divided by 3 equals height in thousands of feet
Write an equation to represent the relationship.
Substitute 45 for d.
15 = h
The plane was flying at 15,000 ft when the descent began.

15. Properties of Equality
WORDS
Addition Property of Equality
You can add the same number to both sides of an equation, and the statement will still be true.
NUMBERS
3 = 3
3 + 2 = 3 + 2
5 = 5
ALGEBRA
a = b
a + c = b + c

16. Properties of Equality
WORDS
Subtraction Property of Equality
You can subtract the same number from both sides of an equation, and the statement will still be true.
NUMBERS
7 = 7
7 – 5 = 7 – 5
2 = 2
ALGEBRA
a = b
a – c = b – c

17. Properties of Equality
WORDS
Multiplication Property of Equality
You can multiply both sides of an equation by the same number, and the statement will still be true.
NUMBERS
6 = 6
6(3) = 6(3)
18 = 18
ALGEBRA
a = b
ac = bc

18. Properties of Equality
Division Property of Equality
You can divide both sides of an equation by the same nonzero number, and the statement will still be true.
WORDS
a = b
(c ≠ 0)
8 = 8
2 = 2
ALGEBRA
NUMBERS 8 4 = a c

19. Lesson Quiz: Part 1 1. 2. 21 3. 8y = 4 4. 126 = 9q 2.8 5. 6. –14 40
Solve each equation. 1. 2.
3. 8y = 4
= 9q 5. 6. 21
2.8
–14 40

20. Lesson Quiz: Part 2
7. A person's weight on Venus is about his or her weight on Earth. Write and solve an equation to find how much a person weighs on Earth if he or she weighs 108 pounds on Venus. 9 10

21. Objectives Solve one-step inequalities by using multiplication.
Solve one-step inequalities by using division.

22. Remember, solving inequalities is similar to solving equations
Remember, solving inequalities is similar to solving equations. To solve an inequality that contains multiplication or division, undo the operation by dividing or multiplying both sides of the inequality by the same number.
The following rules show the properties of inequality for multiplying or dividing by a positive number. The rules for multiplying or dividing by a negative number appear later in this lesson.

23. Solve the inequality and graph the solutions.
Example 1A: Multiplying or Dividing by a Positive Number
Solve the inequality and graph the solutions.
7x > –42
7x > –42
>
Since x is multiplied by 7, divide both sides by 7 to undo the multiplication.
1x > –6
x > –6
–10 –8 –6 –4 –2 2 4 6 8 10

24. Solve the inequality and graph the solutions.
Example 1C: Multiplying or Dividing by a Positive Number
Solve the inequality and graph the solutions.
Since r is multiplied by , multiply both sides by the reciprocal of .
r < 16 2 4 6 8 10 12 14 16 18 20

25. Solve the inequality and graph the solutions.
Check It Out! Example 1a
Solve the inequality and graph the solutions.
4k > 24
Since k is multiplied by 4, divide both sides by 4.
k > 6 2 4 6 8 10 12 16 18 20 14

26. If you multiply or divide both sides of an inequality by a negative number, the resulting inequality is not a true statement. You need to reverse the inequality symbol to make the statement true.

28. Caution!
Do not change the direction of the inequality symbol just because you see a negative sign. For example, you do not change the symbol when solving 4x < –24.

29. Solve the inequality and graph the solutions.
Example 2A: Multiplying or Dividing by a Negative Number
Solve the inequality and graph the solutions.
–12x > 84
Since x is multiplied by –12, divide both sides by –12. Change > to <.
x < –7
–10 –8 –6 –4 –2 2 4 6
–12
–14 –7

30. Solve the inequality and graph the solutions.
Example 2B: Multiplying or Dividing by a Negative Number
Solve the inequality and graph the solutions.
Since x is divided by –3, multiply both sides by –3. Change to .
24  x
(or x  24) 16 18 20 22 24 10 14 26 28 30 12

31. Let p represent the number of tubes of paint that Jill can buy.
Example 3: Application
Jill has a $20 gift card to an art supply store where 4 oz tubes of paint are $4.30 each after tax. What are the possible numbers of tubes that Jill can buy?
Let p represent the number of tubes of paint that Jill can buy.
$4.30
times
number of tubes
is at most
$20.00.
4.30 • p ≤
20.00

32. Example 3 Continued
4.30p ≤ 20.00
Since p is multiplied by 4.30, divide both sides by The symbol does not change.
p ≤ 4.65…
Since Jill can buy only whole numbers of tubes, she can buy 0, 1, 2, 3, or 4 tubes of paint.

33. Solve each inequality and graph the solutions.
Lesson Quiz
Solve each inequality and graph the solutions.
1. 8x < –24
x < –3
2. –5x ≥ 30
x ≤ –6 3.
x > 20 4.
x ≥ 6
5. A soccer coach plans to order more shirts for her team. Each shirt costs $9.85. She has $77 left in her uniform budget. What are the possible number of shirts she can buy?
0, 1, 2, 3, 4, 5, 6, or 7 shirts