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Math Coach Amy
Amy Ferguson Moncure

Algebra 1 · Unit 1

Clearing Fractions (Advanced Strategy)

Multiply every term by the LCD so a linear equation with fractions becomes an integer equation, then solve and check.

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Key Ideas Easy Medium Hard Super Challenge
What to Remember

The old-school way

Solve. y − 25 = −13

y − 25 = −13

Add 25 to both sides.

y = −13 + 25

The LCD of 3 and 5 is 15.

−13 = −515

25 = 615

y = −515 + 615

y = 115

Answer: y = 115

Multiply by the LCD

If there are parentheses, distribute first. Then multiply every term by the LCD, including whole-number terms. That writes an equivalent equation with integers.

Same equation.

y − 25 = −13

The LCD of 5 and 3 is 15. Multiply every term by 15.

15y − 15 × 25 = 15 × (−13)

15y − 6 = −5

Add 6 to both sides.

15y = 1

Divide both sides by 15.

y = 115

Answer: y = 115

Easy example

Solve. 12x + 5 = 9

12x + 5 = 9

The LCD is 2. Multiply every term by 2.

x + 10 = 18

Subtract 10 from both sides.

x = 8

Check.

12(8) + 5 = 4 + 5 = 9

Answer: x = 8

Medium example

Solve. x2 + x5 = 7

x2 + x5 = 7

The LCD of 2 and 5 is 10. Multiply every term by 10.

5x + 2x = 70

7x = 70

x = 10

Check.

102 + 105 = 5 + 2 = 7

Answer: x = 10

Hard example

Solve. 23(x + 3) = 12(x + 8)

23(x + 3) = 12(x + 8)

Distribute first.

23x + 2 = 12x + 4

The LCD of 3 and 2 is 6. Multiply every term by 6.

4x + 12 = 3x + 24

Subtract 3x from both sides.

x + 12 = 24

Subtract 12 from both sides.

x = 12

Check.

23(12 + 3) = 23(15) = 10

12(12 + 8) = 12(20) = 10

Answer: x = 12

Level 1

Easy

One denominator. Multiply every term by that number.

Example
Solve. 12x + 5 = 9

12x + 5 = 9

The LCD is 2. Multiply every term by 2.

x + 10 = 18

Subtract 10 from both sides.

x = 8

Check.

12(8) + 5 = 4 + 5 = 9

Answer: x = 8

1
Solve. 13x − 2 = 4
Show Answer

13x − 2 = 4

The LCD is 3. Multiply every term by 3.

x − 6 = 12

Add 6 to both sides.

x = 18

Check.

13(18) − 2 = 6 − 2 = 4

Answer: x = 18

2
Solve. x5 + 1 = 6
Show Answer

x5 + 1 = 6

The LCD is 5. Multiply every term by 5.

x + 5 = 30

Subtract 5 from both sides.

x = 25

Check.

255 + 1 = 5 + 1 = 6

Answer: x = 25

3
Solve. 4 = x2 − 3
Show Answer

4 = x2 − 3

The LCD is 2. Multiply every term by 2.

8 = x − 6

Add 6 to both sides.

x = 14

Check.

142 − 3 = 7 − 3 = 4

Answer: x = 14

4
Solve. 34x = 9
Show Answer

34x = 9

The LCD is 4. Multiply both sides by 4.

3x = 36

Divide both sides by 3.

x = 12

Check.

34(12) = 9

Answer: x = 12

5
Solve. 12x + 12 = 7
Show Answer

12x + 12 = 7

The LCD is 2. Multiply every term by 2.

x + 1 = 14

Subtract 1 from both sides.

x = 13

Check.

12(13) + 12 = 132 + 12 = 7

Answer: x = 13

Level 2

Medium

Two different denominators. Use the LCD of both.

Example
Solve. x2 + x5 = 7

x2 + x5 = 7

The LCD of 2 and 5 is 10. Multiply every term by 10.

5x + 2x = 70

7x = 70

x = 10

Check.

102 + 105 = 5 + 2 = 7

Answer: x = 10

1
Solve. x3 + x6 = 4
Show Answer

x3 + x6 = 4

The LCD of 3 and 6 is 6. Multiply every term by 6.

2x + x = 24

3x = 24

x = 8

Check.

83 + 86 = 83 + 43 = 4

Answer: x = 8

2
Solve. 14x + 3 = 13x − 1
Show Answer

14x + 3 = 13x − 1

The LCD of 4 and 3 is 12. Multiply every term by 12.

3x + 36 = 4x − 12

Subtract 3x from both sides.

36 = x − 12

Add 12 to both sides.

x = 48

Check.

14(48) + 3 = 12 + 3 = 15

13(48) − 1 = 16 − 1 = 15

Answer: x = 48

3
Solve. x6 + 2 = x3
Show Answer

x6 + 2 = x3

The LCD of 6 and 3 is 6. Multiply every term by 6.

x + 12 = 2x

Subtract x from both sides.

12 = x

Check.

126 + 2 = 2 + 2 = 4

123 = 4

Answer: x = 12

4
Solve. 34 + x2 = 54
Show Answer

34 + x2 = 54

The LCD of 4 and 2 is 4. Multiply every term by 4.

3 + 2x = 5

Subtract 3 from both sides.

2x = 2

x = 1

Check.

34 + 12 = 34 + 24 = 54

Answer: x = 1

5
Solve. 56x − 2 = 13x + 4
Show Answer

56x − 2 = 13x + 4

The LCD of 6 and 3 is 6. Multiply every term by 6.

5x − 12 = 2x + 24

Subtract 2x from both sides.

3x − 12 = 24

Add 12 to both sides.

3x = 36

x = 12

Check.

56(12) − 2 = 10 − 2 = 8

13(12) + 4 = 4 + 4 = 8

Answer: x = 12

Level 3

Hard

If there are parentheses, distribute first. Then clear the fractions.

Example
Solve. 23(x + 3) = 12(x + 8)

23(x + 3) = 12(x + 8)

Distribute first.

23x + 2 = 12x + 4

The LCD of 3 and 2 is 6. Multiply every term by 6.

4x + 12 = 3x + 24

Subtract 3x from both sides.

x + 12 = 24

Subtract 12 from both sides.

x = 12

Check.

23(12 + 3) = 23(15) = 10

12(12 + 8) = 12(20) = 10

Answer: x = 12

1
Solve. 13(2x − 1) = 14x + 12
Show Answer

13(2x − 1) = 14x + 12

Distribute first.

23x − 13 = 14x + 12

The LCD of 3, 4, and 2 is 12. Multiply every term by 12.

8x − 4 = 3x + 6

Subtract 3x from both sides.

5x − 4 = 6

Add 4 to both sides.

5x = 10

x = 2

Check.

13(4 − 1) = 1

14(2) + 12 = 12 + 12 = 1

Answer: x = 2

2
Solve. x4 + 23 = x3 − 16
Show Answer

x4 + 23 = x3 − 16

The LCD of 4, 3, and 6 is 12. Multiply every term by 12.

3x + 8 = 4x − 2

Add 2 to both sides.

3x + 10 = 4x

Subtract 3x from both sides.

10 = x

Check.

104 + 23 = 52 + 23 = 156 + 46 = 196

103 − 16 = 206 − 16 = 196

Answer: x = 10

3
Solve. 56x + 12 = 23x + 53
Show Answer

56x + 12 = 23x + 53

The LCD of 6, 2, and 3 is 6. Multiply every term by 6.

5x + 3 = 4x + 10

Subtract 4x from both sides.

x + 3 = 10

Subtract 3 from both sides.

x = 7

Check.

56(7) + 12 = 356 + 36 = 386 = 193

23(7) + 53 = 143 + 53 = 193

Answer: x = 7

4
Solve. 2x5 − x4 = 1120
Show Answer

2x5 − x4 = 1120

The LCD of 5, 4, and 20 is 20. Multiply every term by 20.

8x − 5x = 11

3x = 11

x = 113

Check.

25×113 − 14×113 = 2215 − 1112

8860 − 5560 = 3360 = 1120

Answer: x = 113

5
Solve. 34x − 12 = 18x + 1
Show Answer

34x − 12 = 18x + 1

The LCD of 4, 2, and 8 is 8. Multiply every term by 8.

6x − 4 = x + 8

Subtract x from both sides.

5x − 4 = 8

Add 4 to both sides.

5x = 12

x = 125

Check.

34×125 − 12 = 95 − 12 = 1810 − 510 = 1310

18×125 + 1 = 1240 + 1 = 310 + 1 = 1310

Answer: x = 125

Super Challenge

Super Challenge

1
Solve. x4 + x6 + x8 = 138
Show Answer

x4 + x6 + x8 = 138

The LCD of 4, 6, and 8 is 24. Multiply every term by 24.

6x + 4x + 3x = 39

13x = 39

x = 3

Check.

34 + 36 + 38 = 68 + 48 + 38 = 138

Answer: x = 3

2
Write 0.6x + 0.3 = 0.2x + 1.5 with fractions, then clear the denominators and solve.
Show Answer

0.6 = 35, 0.3 = 310, 0.2 = 15, 1.5 = 32.

35x + 310 = 15x + 32

The LCD of 5, 10, and 2 is 10. Multiply every term by 10.

6x + 3 = 2x + 15

Subtract 2x from both sides.

4x + 3 = 15

Subtract 3 from both sides.

4x = 12

x = 3

Check.

0.6(3) + 0.3 = 1.8 + 0.3 = 2.1

0.2(3) + 1.5 = 0.6 + 1.5 = 2.1

Answer: x = 3

3
Write 0.25x + 0.5 = 0.1x + 1.1 with fractions, then clear the denominators and solve.
Show Answer

0.25 = 14, 0.5 = 12, 0.1 = 110, 1.1 = 1110.

14x + 12 = 110x + 1110

The LCD of 4, 2, and 10 is 20. Multiply every term by 20.

5x + 10 = 2x + 22

Subtract 2x from both sides.

3x + 10 = 22

Subtract 10 from both sides.

3x = 12

x = 4

Check.

0.25(4) + 0.5 = 1 + 0.5 = 1.5

0.1(4) + 1.1 = 0.4 + 1.1 = 1.5

Answer: x = 4

4
Solve. 14(x + 12) − 16(x + 12) = 2
Show Answer

14(x + 12) − 16(x + 12) = 2

Distribute first.

14x + 3 − 16x − 2 = 2

The LCD of 4 and 6 is 12. Multiply every term by 12.

3x + 36 − 2x − 24 = 24

x + 12 = 24

Subtract 12 from both sides.

x = 12

Check.

14(12 + 12) − 16(12 + 12) = 14(24) − 16(24) = 6 − 4 = 2

Answer: x = 12

5
Solve. 12(x − 10) − 15(x − 10) = 3
Show Answer

12(x − 10) − 15(x − 10) = 3

Distribute first.

12x − 5 − 15x + 2 = 3

The LCD of 2 and 5 is 10. Multiply every term by 10.

5x − 50 − 2x + 20 = 30

3x − 30 = 30

Add 30 to both sides.

3x = 60

x = 20

Check.

12(20 − 10) − 15(20 − 10) = 12(10) − 15(10) = 5 − 2 = 3

Answer: x = 20

6
Solve. 15(x + 15) − 13(x + 15) = −4
Show Answer

15(x + 15) − 13(x + 15) = −4

Distribute first.

15x + 3 − 13x − 5 = −4

The LCD of 5 and 3 is 15. Multiply every term by 15.

3x + 45 − 5x − 75 = −60

−2x − 30 = −60

Add 30 to both sides.

−2x = −30

Divide both sides by −2.

x = 15

Check.

15(15 + 15) − 13(15 + 15) = 15(30) − 13(30) = 6 − 10 = −4

Answer: x = 15

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