Algebra Foundations for Normal People by Gabriel Beguerie - HTML preview

Chapter 7

Worked example (no skipped steps)

7.1 Solve 3( 𝑥 + 2) = 18.

Distribute:

3 𝑥 + 6 = 18

Subtract 6:

3 𝑥 = 12

Divide by 3:

𝑥 = 4

160

Step-by-step solutions

Worked example (no skipped steps)

7.2 Solve 4(2 𝑦 − 1) = 20.

Distribute:

8 𝑦 − 4 = 20

Add 4:

8 𝑦 = 24

Divide by 8:

𝑦 = 3

Worked example (no skipped steps)

7.3 Solve 5 − 2( 𝑥 + 1) = −9.

Distribute:

5 − 2 𝑥 − 2 = −9

Combine like terms:

3 − 2 𝑥 = −9

Subtract 3:

−2 𝑥 = −12

Divide by −2:

𝑥 = 6

Worked example (no skipped steps)

7.4 Solve 2( 𝑥 − 3) + 4 = 10.

Distribute:

2 𝑥 − 6 + 4 = 10

Combine:

2 𝑥 − 2 = 10

Add 2:

2 𝑥 = 12

Divide by 2:

𝑥 = 6

Chapter 7

161

Worked example (no skipped steps)

7.5 Solve 5 𝑥 − 7 = 2 𝑥 + 8.

Subtract 2 𝑥:

3 𝑥 − 7 = 8

Add 7:

3 𝑥 = 15

Divide by 3:

𝑥 = 5

Worked example (no skipped steps)

7.6 Solve 6 𝑎 + 3 = 2 𝑎 + 15.

Subtract 2 𝑎:

4 𝑎 + 3 = 15

Subtract 3:

4 𝑎 = 12

Divide by 4:

𝑎 = 3

Worked example (no skipped steps)

7.7 Solve 3 𝑥 + 4 = 2 𝑥 + 11.

Subtract 2 𝑥:

𝑥 + 4 = 11

Subtract 4:

𝑥 = 7

Worked example (no skipped steps)

7.8 Solve 4( 𝑥 − 1) + 3 = 2 𝑥 + 9.

Distribute:

4 𝑥 − 4 + 3 = 2 𝑥 + 9

Combine:

4 𝑥 − 1 = 2 𝑥 + 9

Subtract 2 𝑥:

2 𝑥 − 1 = 9

Add 1:

2 𝑥 = 10

Divide by 2:

𝑥 = 5

162

Step-by-step solutions

Worked example (no skipped steps)

7.9 Solve 2( 𝑥 + 3) − 4 = 𝑥 + 9.

Distribute:

2 𝑥 + 6 − 4 = 𝑥 + 9

Combine:

2 𝑥 + 2 = 𝑥 + 9

Subtract 𝑥:

𝑥 + 2 = 9

Subtract 2:

𝑥 = 7

Worked example (no skipped steps)

7.10 Solve 3 − (2 𝑥 − 5) = 12.

Distribute the negative sign:

3 − 2 𝑥 + 5 = 12

Combine:

8 − 2 𝑥 = 12

Subtract 8:

−2 𝑥 = 4

Divide by −2:

𝑥 = −2

Worked example (no skipped steps)

7.11 Solve 𝑥 + 𝑥 = 5.

2

3

Multiply every term by 6:

𝑥

𝑥

6 ·

+ 6 ·

= 6 · 5

2

3

Simplify:

3 𝑥 + 2 𝑥 = 30

So:

5 𝑥 = 30

Divide by 5:

𝑥 = 6

Chapter 7

163

Worked example (no skipped steps)

7.12 Solve 𝑥 + 5 = 9.

3

Subtract 5:

𝑥 = 4

3

Multiply by 3:

𝑥 = 12

Worked example (no skipped steps)

7.13 Solve 2 𝑥−1 = 5.

3

Multiply by 3:

2 𝑥 − 1 = 15

Add 1:

2 𝑥 = 16

Divide by 2:

𝑥 = 8

Worked example (no skipped steps)

7.14 Solve 𝑥+4 = 7.

2

Multiply by 2:

𝑥 + 4 = 14

Subtract 4:

𝑥 = 10

Worked example (no skipped steps)

7.15 Solve 2 𝑥 + 5 = 2 𝑥 − 1.

Subtract 2 𝑥 from both sides:

5 = −1

This is false, so the equation has no solution.

Worked example (no skipped steps)

7.16 Solve 4( 𝑥 + 1) = 4 𝑥 + 4.

Distribute:

4 𝑥 + 4 = 4 𝑥 + 4

Subtract 4 𝑥:

4 = 4

This is always true, so the equation has infinitely many solutions.

164

Step-by-step solutions

Worked example (no skipped steps)

7.17 Solve 3( 𝑥 − 2) = 3 𝑥 + 5.

Distribute:

3 𝑥 − 6 = 3 𝑥 + 5

Subtract 3 𝑥:

−6 = 5

This is false, so there is no solution.

Worked example (no skipped steps)

7.18 Solve 7 − 2( 𝑦 + 3) = 1.

Distribute:

7 − 2 𝑦 − 6 = 1

Combine:

1 − 2 𝑦 = 1

Subtract 1:

−2 𝑦 = 0

Divide by −2:

𝑦 = 0

Worked example (no skipped steps)

7.19 Solve 5( 𝑥 − 2) = 3 𝑥 + 6.

Distribute:

5 𝑥 − 10 = 3 𝑥 + 6

Subtract 3 𝑥:

2 𝑥 − 10 = 6

Add 10:

2 𝑥 = 16

Divide by 2:

𝑥 = 8

Worked example (no skipped steps)

7.20 Solve 3 𝑥+6 = 8.

3

Multiply by 3:

3 𝑥 + 6 = 24

Subtract 6:

3 𝑥 = 18

Divide by 3:

𝑥 = 6

Chapter 7

165

Worked example (no skipped steps)

7.21 Solve 2(3 𝑚 − 1) + 5 = 15.

Distribute:

6 𝑚 − 2 + 5 = 15

Combine:

6 𝑚 + 3 = 15

Subtract 3:

6 𝑚 = 12

Divide by 6:

𝑚 = 2

Worked example (no skipped steps)

7.22 Solve 4 − 3(2 𝑥 − 1) = −11.

Distribute:

4 − 6 𝑥 + 3 = −11

Combine:

7 − 6 𝑥 = −11

Subtract 7:

−6 𝑥 = −18

Divide by −6:

𝑥 = 3

Worked example (no skipped steps)

7.23 Decide whether 4 𝑎 + 7 = 4 𝑎 − 2 has one solution, no solution, or infinitely many solutions.

Subtract 4 𝑎 from both sides:

7 = −2

This is false, so there is no solution.

166

Step-by-step solutions

Worked example (no skipped steps)

7.24 Decide whether 3( 𝑏 − 2) = 3 𝑏 − 6 has one solution, no solution, or infinitely many solutions.

Distribute:

3 𝑏 − 6 = 3 𝑏 − 6

Subtract 3 𝑏:

−6 = −6

This is always true, so there are infinitely many solutions.