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Course 3
11-2 Solving Multi-Step Equations
To solve a multi-step equation, you may have to simplify the equation first by combining like terms.
Course 3
11-2 Solving Multi-Step Equations
Solve.
8x + 6 + 3x – 2 = 37
Additional Example 1: Solving Equations That Contain Like Terms
11x + 4 = 37 Combine like terms. – 4 – 4 Subtract 4 from both sides.
11x = 33
x = 3
Divide both sides by 11.3311
11x11
=
Course 3
11-2 Solving Multi-Step Equations
Check
Additional Example 1 Continued
8x + 6 + 3x – 2 = 37
8(3) + 6 + 3(3) – 2 = 37?
24 + 6 + 9 – 2 = 37?
37 = 37?
Substitute 3 for x.
Course 3
11-2 Solving Multi-Step Equations
Solve.
9x + 5 + 4x – 2 = 42
Check It Out: Example 1
13x + 3 = 42 Combine like terms.
– 3 – 3 Subtract 3 from both sides.13x = 39
x = 3
Divide both sides by 13.3913
13x13
=
Course 3
11-2 Solving Multi-Step Equations
Check
Check It Out: Example 1 Continued
9x + 5 + 4x – 2 = 42
9(3) + 5 + 4(3) – 2 = 42?
27 + 5 + 12 – 2 = 42 ?
42 = 42?
Substitute 3 for x.
Course 3
11-2 Solving Multi-Step Equations
Solve.
+ = –
Check It Out: Example 2A
1 4
5 4
3n4
Multiply both sides by 4 to clear fractions, and then solve.
( ) ( )54
–1 4
3n4
4 + = 4
( ) ( ) ( )3n4
54
–1 44 + 4 = 4
3n + 5 = –1
Distributive Property.
Course 3
11-2 Solving Multi-Step Equations
Check It Out: Example 2A Continued
3n + 5 = –1 – 5 –5 Subtract 5 from both sides.
3n = –6
3n3
–6 3
= Divide both sides by 3.
n = –2
Course 3
11-2 Solving Multi-Step Equations
Solve.
1.6x + 3x – x + 9 = 33
2. + =
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?
Lesson Quiz
x = 3
x = 2858
x8
33 8
$8.50
Course 3
11-2 Solving Multi-Step Equations
Answers 1 - 71. x= -2.2 2. w = 2.75
3. x = 11
4. b = -7 5. m = 1
6. x = 25
7. m = 15
Course 3
11-2 Solving Multi-Step Equations
Solve.
4x + 6 = x
Additional Example 1A: Solving Equations with Variables on Both Sides
4x + 6 = x– 4x – 4x
6 = –3x
Subtract 4x from both sides.
Divide both sides by –3.
–2 = x
6–3
–3x–3=
Course 3
11-3 Solving Equations with Variables on Both Sides
Course 3
11-2 Solving Multi-Step Equations
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
Course 3
11-2 Solving Multi-Step Equations
Solve.
9b – 6 = 5b + 18
Additional Example 1B: Solving Equations with Variables on Both Sides
9b – 6 = 5b + 18– 5b – 5b
4b – 6 = 18
4b 4
24 4 =
Subtract 5b from both sides.
Divide both sides by 4.
b = 6
+ 6 + 6
4b = 24Add 6 to both sides.
Course 3
11-3 Solving Equations with Variables on Both Sides
Course 3
11-2 Solving Multi-Step Equations
Solve.
5x + 8 = x
Check It Out: Example 1A
5x + 8 = x– 5x – 5x
8 = –4x
Subtract 5x from both sides.
Divide both sides by –4.
–2 = x
8–4
–4x–4=
Course 3
11-3 Solving Equations with Variables on Both Sides
Course 3
11-2 Solving Multi-Step Equations
Solve.
3b – 2 = 2b + 123b – 2 = 2b + 12
– 2b – 2b
b – 2 = 12
Subtract 2b from both sides.
+ 2 + 2
b = 14Add 2 to both sides.
Check It Out: Example 1B
Course 3
11-3 Solving Equations with Variables on Both Sides
Course 3
11-2 Solving Multi-Step Equations
Solve.
10z – 15 – 4z = 8 – 2z - 15
Additional Example 2A: Solving Multi-Step Equations with Variables on Both Sides
10z – 15 – 4z = 8 – 2z – 15
+ 15 +15
6z – 15 = –2z – 7 Combine like terms.+ 2z + 2z Add 2z to both sides.
8z – 15 = – 7
8z = 8
z = 1
Add 15 to both sides.
Divide both sides by 8.8z 88 8=
Course 3
11-3 Solving Equations with Variables on Both Sides
Course 3
11-2 Solving Multi-Step Equations
Solve.
12z – 12 – 4z = 6 – 2z + 32
Check It Out: Example 2A
12z – 12 – 4z = 6 – 2z + 32
+ 12 +12
8z – 12 = –2z + 38 Combine like terms.+ 2z + 2z Add 2z to both sides.
10z – 12 = 38
10z = 50
z = 5
Add 12 to both sides.
Divide both sides by 10.10z 5010 10=
Course 3
11-3 Solving Equations with Variables on Both Sides
Course 3
11-2 Solving Multi-Step Equations
Subtract 18 from both sides.
2y + 18 = – 18
2y = –36
– 18 – 18
–36 2
2y2 = Divide both sides by 2.
y = –18
26y + 18 = 24y – 18
– 24y – 24y Subtract 24y from both sides.
Check It Out: Example 2B Continued
Course 3
11-3 Solving Equations with Variables on Both Sides
Course 3
11-2 Solving Multi-Step Equations
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.
Course 3
11-3 Solving Equations with Variables on Both Sides
Course 3
11-2 Solving Multi-Step Equations
Additional Example 3 Continued
39.95 + 2.95r = 26.00 + 4.50rLet r represent the price of one rose.
– 2.95r – 2.95r
39.95 = 26.00 + 1.55r
Subtract 2.95r from both sides.
– 26.00 – 26.00 Subtract 26.00 from both sides.
13.95 = 1.55r 13.951.55
1.55r 1.55= Divide both sides by 1.55.
9 = r
The two services would cost the same when using 9 roses.Course 3
11-3 Solving Equations with Variables on Both Sides
Course 3
11-2 Solving Multi-Step Equations
Lesson Quiz
Solve.
1. 4x + 16 = 2x
2. 8x – 3 = 15 + 5x
3. x = x – 9
x = 6
x = –8
Insert Lesson Title Here
x = 3614
12
Course 3
11-3 Solving Equations with Variables on Both Sides
Course 3
11-2 Solving Multi-Step Equations
Answers 1 - 9
1. x = 9 2. k = 10.2
3. d = 2 4. a = -4
5. x = 3 6. d = ¾
7. x = 7 8. y = -2/5
9. x = 4
Course 3
11-2 Solving Multi-Step Equations
Warm UpSolve.
1. 6x + 36 = 2x
2. 4x – 13 = 15 + 5x
3. 5(x – 3) = 2x + 3
x = –9
x = –28
x = 6
Course 3
11-5 Solving Two-Step Inequalities
Course 3
11-2 Solving Multi-Step Equations
Solve and graph.
Additional Example 1A: Solving Two-Step Inequalities
4x + 1 > 13
4x + 1 > 13 – 1 – 1 Subtract 1 from both sides.
4x > 124x4
> 124
Divide both sides by 4.
x > 3 1 2 3 4 5 6 7
Course 3
11-5 Solving Two-Step Inequalities
Course 3
11-2 Solving Multi-Step Equations
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!
Course 3
11-2 Solving Multi-Step Equations
Additional Example 1B: Solving Two-Step Inequalities
–9x + 7 25
–9x + 7 25
– 7 – 7 Subtract 7 from both sides.
–9x 18
–9x–9
18–9
Divide each side by –9; change to .
x –2-6 -5 -4 -3 -2 -1 0
Course 3
11-5 Solving Two-Step Inequalities
Solve and graph.
Course 3
11-2 Solving Multi-Step Equations
Solve and graph.
Check It Out: Example 1A
5x + 2 > 12
5x + 2 > 12 – 2 – 2 Subtract 2 from both sides.
5x > 105x5
> 105
Divide both sides by 5.
x > 2 1 2 3 4 5 6 7
Course 3
11-5 Solving Two-Step Inequalities
Course 3
11-2 Solving Multi-Step Equations
–4x + 2 18
–4x + 2 18
– 2 – 2 Subtract 2 from both sides.
–4x 16
–4x–4
16–4
Divide each side by –4; change to .
x –4-6 -5 -4 -3 -2 -1 0
Check It Out: Example 1B
Course 3
11-5 Solving Two-Step Inequalities
Course 3
11-2 Solving Multi-Step Equations
Check your Understanding
Solve and graph.
1. 4x – 6 > 10
2. 7x + 9 < 3x – 15
3. w – 3w < 32
x < –6
x > 4
Insert Lesson Title Here
w > –16
1 2 3 4 5 6 7
-10 -9 -8 -7 -6 -5 -4
-18 -17 -16 -15 -14 -13 -12
Course 3
11-5 Solving Two-Step Inequalities
Course 3
11-2 Solving Multi-Step Equations
1. 2.
3. 4.
5. 6.
7. 9.