Algebra Foundations for Normal People by Gabriel Beguerie - HTML preview

Chapter 5

Worked example (no skipped steps)

5.1 Simplify 5 𝑥 + 2 𝑥.

5 𝑥 + 2 𝑥 = 7 𝑥

Worked example (no skipped steps)

5.2 Simplify 7 𝑎 − 3 + 2 𝑎 + 8.

Combine like terms:

7 𝑎 + 2 𝑎 = 9 𝑎,

−3 + 8 = 5

So:

9 𝑎 + 5

152

Step-by-step solutions

Worked example (no skipped steps)

5.3 Expand 4( 𝑥 + 3).

4 𝑥 + 12

Worked example (no skipped steps)

5.4 Expand −3( 𝑦 − 2).

−3 𝑦 + 6

Worked example (no skipped steps)

5.5 Expand and simplify 2(3 𝑥 + 1) − 4.

2(3 𝑥 + 1) − 4 = 6 𝑥 + 2 − 4 = 6 𝑥 − 2

Worked example (no skipped steps)

5.6 Expand and simplify −2(4 𝑥 − 5) + 3.

−8 𝑥 + 10 + 3 = −8 𝑥 + 13

Worked example (no skipped steps)

5.7 Decide whether 3 𝑥 and 3 are like terms.

No. One has a variable and one does not.

Worked example (no skipped steps)

5.8 Simplify 6 𝑚 − 2 𝑚 + 9 − 4.

6 𝑚 − 2 𝑚 = 4 𝑚,

9 − 4 = 5

So the result is

4 𝑚 + 5

Worked example (no skipped steps)

5.9 Simplify 4 𝑥 + 7 − 2 𝑥 + 3.

Combine the like terms with 𝑥:

4 𝑥 − 2 𝑥 = 2 𝑥

Combine the constants:

7 + 3 = 10

So the expression becomes:

2 𝑥 + 10

Chapter 5

153

Answer: 2 𝑥 + 10.

Worked example (no skipped steps)

5.10 Expand 3( 𝑥 − 5).

Distribute the 3 to each term inside the parentheses:

3( 𝑥 − 5) = 3 · 𝑥 + 3 · (−5)

So:

3( 𝑥 − 5) = 3 𝑥 − 15

Answer: 3 𝑥 − 15.

Worked example (no skipped steps)

5.11 Simplify −2(4 𝑦 − 1) + 3 𝑦.

Distribute −2 first:

−2(4 𝑦 − 1) = −8 𝑦 + 2

Now add the remaining 3 𝑦:

−8 𝑦 + 2 + 3 𝑦

Combine the like terms:

−8 𝑦 + 3 𝑦 = −5 𝑦

So the result is:

−5 𝑦 + 2

Answer: −5 𝑦 + 2.

Worked example (no skipped steps)

5.12 Evaluate 2 𝑎 + 5 𝑏 when 𝑎 = 3 and 𝑏 = −1.

Substitute the values:

2 𝑎 + 5 𝑏 = 2(3) + 5(−1)

Multiply:

2(3) = 6 ,

5(−1) = −5

Now add:

6 + (−5) = 1

Answer: 1.

154

Step-by-step solutions

Worked example (no skipped steps)

5.13 Simplify 3 𝑥 + 5 − 2 𝑥 + 7.

Combine like terms:

3 𝑥 − 2 𝑥 = 𝑥

5 + 7 = 12

So:

𝑥 + 12

Answer: 𝑥 + 12.

Worked example (no skipped steps)

5.14 Distribute and simplify 4( 𝑥 − 3).

4( 𝑥 − 3) = 4 𝑥 − 12

Answer: 4 𝑥 − 12.

Worked example (no skipped steps)

5.15 Distribute and simplify −2(5 𝑦 + 1).

−2(5 𝑦 + 1) = −10 𝑦 − 2

Answer: −10 𝑦 − 2.

Worked example (no skipped steps)

5.16 Simplify 2(3 𝑎 − 4) + 5 𝑎.

Distribute first:

2(3 𝑎 − 4) = 6 𝑎 − 8

Then combine like terms:

6 𝑎 − 8 + 5 𝑎 = 11 𝑎 − 8

Answer: 11 𝑎 − 8.

Worked example (no skipped steps)

5.17 Simplify 7 − (2 𝑥 − 5).

Distribute the negative sign:

7 − 2 𝑥 + 5

Combine constants:

12 − 2 𝑥

Answer: 12 − 2 𝑥.

Chapter 6

155

Worked example (no skipped steps)

5.18 Combine like terms: 6 𝑚 − 2 + 4 𝑚 + 9.

6 𝑚 + 4 𝑚 = 10 𝑚

−2 + 9 = 7

Answer: 10 𝑚 + 7.

Worked example (no skipped steps)

5.19 Simplify 3(2 𝑝 + 5) − 4( 𝑝 − 1).

Distribute:

6 𝑝 + 15 − 4 𝑝 + 4

Combine like terms:

2 𝑝 + 19

Answer: 2 𝑝 + 19.

Worked example (no skipped steps)

5.20 Simplify 5 − 3( 𝑥 − 2).

Distribute the −3:

5 − 3 𝑥 + 6 = 11 − 3 𝑥

Answer: 11 − 3 𝑥.