Algebra Foundations for Normal People by Gabriel Beguerie - HTML preview

Chapter 1

Worked example (no skipped steps)

1.1 If 𝑥 = 4, evaluate 𝑥 + 3.

Substitute 4 for 𝑥:

4 + 3 = 7

Answer: 7.

Worked example (no skipped steps)

1.2 If 𝑥 = −2, evaluate 2 𝑥 + 1.

2(−2) + 1 = −4 + 1 = −3

Answer: −3.

Worked example (no skipped steps)

1.3 Simplify 5 𝑎 + 3 𝑎.

Like terms combine:

5 𝑎 + 3 𝑎 = 8 𝑎

Worked example (no skipped steps)

1.4 Simplify 7 𝑦 − 𝑦.

7 𝑦 − 𝑦 = 6 𝑦

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Step-by-step solutions

Worked example (no skipped steps)

1.5 Decide whether 3 𝑥 + 2 is an expression or an equation.

There is no equal sign, so it is an expression.

Worked example (no skipped steps)

1.6 Decide whether 3 𝑥 + 2 = 11 is an expression or an equation.

There is an equal sign, so it is an equation.

Worked example (no skipped steps)

1.7 Solve 𝑚 + 5 = 12.

Subtract 5 from both sides:

𝑚 = 7

Worked example (no skipped steps)

1.8 Solve 𝑝 − 4 = 9.

Add 4 to both sides:

𝑝 = 13

Worked example (no skipped steps)

1.9 If 𝑎 = 3, evaluate 4 𝑎 − 7.

Substitute 3 for 𝑎:

4 𝑎 − 7 = 4(3) − 7

Multiply first:

4(3) = 12

Then subtract:

12 − 7 = 5

Answer: 5.

Worked example (no skipped steps)

1.10 If 𝑏 = −5, evaluate 2 𝑏 + 6.

Substitute −5 for 𝑏:

2 𝑏 + 6 = 2(−5) + 6

Multiply:

2(−5) = −10

Then add:

−10 + 6 = −4

Answer: −4.

Chapter 1

135

Worked example (no skipped steps)

1.11 Simplify 9 𝑚 − 2 𝑚 + 4.

The like terms are 9 𝑚 and −2 𝑚. Combine them: 9 𝑚 − 2 𝑚 = 7 𝑚

The constant 4 stays separate, so:

9 𝑚 − 2 𝑚 + 4 = 7 𝑚 + 4

Answer: 7 𝑚 + 4.

Worked example (no skipped steps)

1.12 Solve 𝑞 − 8 = 6.

Add 8 to both sides:

𝑞 − 8 + 8 = 6 + 8

So:

𝑞 = 14

Check:

14 − 8 = 6

Correct.

Worked example (no skipped steps)

1.13 If 𝑥 = 6, evaluate 3 𝑥 − 4.

Substitute 6 for 𝑥:

3(6) − 4 = 18 − 4 = 14

Answer: 14.

Worked example (no skipped steps)

1.14 If 𝑡 = −3, evaluate 4 𝑡 + 2.

Substitute −3 for 𝑡:

4 𝑡 + 2 = 4(−3) + 2

Multiply first:

4(−3) = −12

Then add:

−12 + 2 = −10

Answer: −10.

136

Step-by-step solutions

Worked example (no skipped steps)

1.15 Simplify 4 𝑝 + 𝑝 − 7.

Combine like terms:

4 𝑝 + 𝑝 = 5 𝑝

So the expression becomes:

5 𝑝 − 7

Answer: 5 𝑝 − 7.

Worked example (no skipped steps)

1.16 Simplify 9 𝑚 − 4 𝑚 + 2.

Combine the 𝑚-terms:

9 𝑚 − 4 𝑚 = 5 𝑚

So:

5 𝑚 + 2

Answer: 5 𝑚 + 2.

Worked example (no skipped steps)

1.17 Solve 𝑛 − 7 = 5.

Add 7 to both sides:

𝑛 = 12

Answer: 𝑛 = 12.

Worked example (no skipped steps)

1.18 Solve 3 𝑘 = 21.

Divide both sides by 3:

𝑘 = 7

Answer: 𝑘 = 7.