1. To solve a multi-step equation, you may have to simplify the equation first by combining like terms.

2. Additional Example 1: Solving Equations That Contain Like Terms
Solve.
8x x – 2 = 37
11x + 4 = 37 Combine like terms.
– 4 – 4 Subtract 4 from both sides.
11x = 33 33 11
11x =
Divide both sides by 11.
x = 3

3. Additional Example 1 Continued
Check
8x x – 2 = 37
8(3) (3) – 2 = 37 ?
Substitute 3 for x.
– 2 = 37 ?
37 = 37 ? 

4. Check It Out: Example 1
Solve.
9x x – 2 = 42
13x + 3 = 42 Combine like terms.
– 3 – 3 Subtract 3 from both sides.
13x = 39 39 13
13x =
Divide both sides by 13.
x = 3

5. Check It Out: Example 1 Continued
9x x – 2 = 42
9(3) (3) – 2 = 42 ?
Substitute 3 for x.
– 2 = 42 ?
42 = 42 ? 

6. Check It Out: Example 2A
Solve.
+ = – 3n 4 5 4 1 4
Multiply both sides by 4 to clear fractions, and then solve.
( ) ( ) 5 4 –1 3n
= 4
( ) ( ) ( ) 3n 4 5 –1
= 4
Distributive Property.
3n + 5 = –1

7. Check It Out: Example 2A Continued
– 5 – Subtract 5 from both sides.
3n = –6 3n 3 –6 =
Divide both sides by 3.
n = –2

8. Lesson Quiz Solve. 6x + 3x – x + 9 = 33 + = x = 3 x = 28
+ =
3. Linda is paid double her normal hourly rate for each hour she works over 40 hours in a week. Last week she worked 52 hours and earned $544. What is her hourly rate?
x = 3 5 8 x 8 33 8
x = 28
$8.50

9. Answers 1 - 7 x= -2.2 2. w = 2.75 x = 11 b = -7 5. m = 1 6. x = 25

10. Additional Example 1A: Solving Equations with Variables on Both Sides
Course 3
11-3
Solving Equations with Variables on Both Sides
Additional Example 1A: Solving Equations with Variables on Both Sides
Solve.
4x + 6 = x
4x + 6 = x
– 4x – 4x
Subtract 4x from both sides.
6 = –3x 6 –3
–3x =
Divide both sides by –3.
–2 = x

11. Solving Equations with Variables on Both Sides
Course 3
11-3
Solving Equations with Variables on Both Sides
Check your solution by substituting the value back into the original equation. For example, 4(-2) + 6 = -2 or -2 = -2.
Helpful Hint

12. Additional Example 1B: Solving Equations with Variables on Both Sides
Course 3
11-3
Solving Equations with Variables on Both Sides
Additional Example 1B: Solving Equations with Variables on Both Sides
Solve.
9b – 6 = 5b + 18
9b – 6 = 5b + 18
– 5b – 5b
Subtract 5b from both sides.
4b – 6 = 18
Add 6 to both sides.
4b = 24 4b 4 24 =
Divide both sides by 4.
b = 6

13. Solving Equations with Variables on Both Sides
Course 3
11-3
Solving Equations with Variables on Both Sides
Check It Out: Example 1A
Solve.
5x + 8 = x
5x + 8 = x
– 5x – 5x
Subtract 5x from both sides.
8 = –4x 8 –4
–4x =
Divide both sides by –4.
–2 = x

14. Solving Equations with Variables on Both Sides
Course 3
11-3
Solving Equations with Variables on Both Sides
Check It Out: Example 1B
Solve.
3b – 2 = 2b + 12
3b – 2 = 2b + 12
– 2b – 2b
Subtract 2b from both sides.
b – 2 =
Add 2 to both sides.
b =

15. Solving Equations with Variables on Both Sides
Course 3
11-3
Solving Equations with Variables on Both Sides
Additional Example 2A: Solving Multi-Step Equations with Variables on Both Sides
Solve.
10z – 15 – 4z = 8 – 2z - 15
10z – 15 – 4z = 8 – 2z – 15
6z – 15 = –2z – 7
Combine like terms.
+ 2z z
Add 2z to both sides.
8z – 15 = – 7
Add 15 to both sides.
8z = 8
8z 8 8 =
Divide both sides by 8.
z = 1

16. Solving Equations with Variables on Both Sides
Course 3
11-3
Solving Equations with Variables on Both Sides
Check It Out: Example 2A
Solve.
12z – 12 – 4z = 6 – 2z + 32
12z – 12 – 4z = 6 – 2z + 32
8z – 12 = –2z + 38
Combine like terms.
+ 2z z
Add 2z to both sides.
10z – 12 =
Add 12 to both sides.
10z = 50
10z 10 =
Divide both sides by 10.
z = 5

17. Check It Out: Example 2B Continued
Course 3
11-3
Solving Equations with Variables on Both Sides
Check It Out: Example 2B Continued
26y + 18 = 24y – 18
– 24y – 24y
Subtract 24y from both sides.
2y + 18 = – 18
– – 18
Subtract 18 from both sides.
2y = –36
–36 2 2y =
Divide both sides by 2.
y = –18

18. Additional Example 3: Business Application
Course 3
11-3
Solving Equations with Variables on Both Sides
Additional Example 3: Business Application
Daisy’s Flowers sell a rose bouquet for $39.95 plus $2.95 for every rose. A competing florist sells a similar bouquet for $26.00 plus $4.50 for every rose. Find the number of roses that would make both florist’s bouquets cost the same price.

19. Additional Example 3 Continued
Course 3
11-3
Solving Equations with Variables on Both Sides
Additional Example 3 Continued
Let r represent the price of one rose.
r = r
Subtract 2.95r from both sides.
– 2.95r – 2.95r
= r
Subtract from both sides.
– – 26.00
= r
13.95
1.55
1.55r 1.55 =
Divide both sides by 1.55.
9 = r
The two services would cost the same when using 9 roses.

20. Insert Lesson Title Here
Course 3
11-3
Solving Equations with Variables on Both Sides
Insert Lesson Title Here
Lesson Quiz
Solve.
1. 4x + 16 = 2x
2. 8x – 3 = x
x = x – 9
x = –8
x = 6 1 4 1 2
x = 36

21. Answers 1 - 9 x = 9 2. k = 10.2 d = 2 4. a = -4 5. x = 3 6. d = ¾
7. x = y = -2/5
9. x = 4

22. Solving Two-Step Inequalities
Course 3
11-5
Solving Two-Step Inequalities
Warm Up
Solve.
1. 6x + 36 = 2x
2. 4x – 13 = x
3. 5(x – 3) = 2x + 3
x = –9
x = –28
x = 6

23. Additional Example 1A: Solving Two-Step Inequalities
Course 3
11-5
Solving Two-Step Inequalities
Additional Example 1A: Solving Two-Step Inequalities
Solve and graph.
4x + 1 > 13
4x + 1 > 13
– 1 – 1 Subtract 1 from both sides.
4x > 12 4x 4
> 12
Divide both sides by 4.
x > 3

24. Solving Two-Step Inequalities
Course 3
11-5
Solving Two-Step Inequalities
If both sides of an inequality are multiplied or divided by a negative number, the inequality symbol must be reversed.
Remember!

25. Additional Example 1B: Solving Two-Step Inequalities
Course 3
11-5
Solving Two-Step Inequalities
Additional Example 1B: Solving Two-Step Inequalities
Solve and graph.
–9x + 7  25
–9x + 7  25
– 7 – 7 Subtract 7 from both sides.
–9x  18
–9x –9  18
Divide each side by –9; change  to .
x  –2

26. Solving Two-Step Inequalities
Course 3
11-5
Solving Two-Step Inequalities
Check It Out: Example 1A
Solve and graph.
5x + 2 > 12
5x + 2 > 12
– 2 – 2 Subtract 2 from both sides.
5x > 10 5x 5
> 10
Divide both sides by 5.
x > 2

27. Solving Two-Step Inequalities
Course 3
11-5
Solving Two-Step Inequalities
Check It Out: Example 1B
–4x + 2  18
–4x + 2  18
– 2 – 2 Subtract 2 from both sides.
–4x  16
–4x –4  16
Divide each side by –4; change  to .
x  –4

28. Check your Understanding
Course 3
11-5
Solving Two-Step Inequalities
Insert Lesson Title Here
Check your Understanding
Solve and graph.
1. 4x – 6 > 10
2. 7x + 9 < 3x – 15
3. w – 3w < 32
x > 4
x < –6
w > –16