Algebra Foundations for Normal People by Gabriel Beguerie - HTML preview

Chapter 14

Worked example (no skipped steps)

14.1 Simplify 4 𝑥 − 3 + 2 𝑥 + 9.

Combine like terms:

4 𝑥 + 2 𝑥 = 6 𝑥,

−3 + 9 = 6

So:

4 𝑥 − 3 + 2 𝑥 + 9 = 6 𝑥 + 6

Likely skill: Chapter 5.

Worked example (no skipped steps)

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

Distribute:

−6 𝑥 + 8 + 5

Combine constants:

−6 𝑥 + 13

Likely skill: Chapter 5.

Worked example (no skipped steps)

14.3 Solve 3 𝑥 + 7 = 22.

Subtract 7:

3 𝑥 = 15

Divide by 3:

𝑥 = 5

Likely skill: Chapter 6.

196

Step-by-step solutions

Worked example (no skipped steps)

14.4 Solve 5( 𝑥 − 1) = 20.

Distribute:

5 𝑥 − 5 = 20

Add 5:

5 𝑥 = 25

Divide by 5:

𝑥 = 5

Likely skill: Chapter 7.

Worked example (no skipped steps)

14.5 Solve −4 𝑥 ≥ 12.

Divide by −4 and flip the inequality:

𝑥 ≤ −3

Likely skill: Chapter 8.

Worked example (no skipped steps)

14.6 Simplify 3 + 1 .

4

8

Common denominator 8:

3

6

=

4

8

So:

6

1

7

+

=

8

8

8

Likely skill: Chapter 3.

Worked example (no skipped steps)

14.7 Simplify 𝑎 3 · 𝑎 4.

Add the exponents:

𝑎 3+4 = 𝑎 7

Likely skill: Chapter 9.

Worked example (no skipped steps)

14.8 Factor 𝑥 2 + 8 𝑥 + 15.

We need two numbers that multiply to 15 and add to 8. Those numbers are 3 and 5.

So:

𝑥 2 + 8 𝑥 + 15 = ( 𝑥 + 3)( 𝑥 + 5) Likely skill: Chapter 10.

Chapter 14

197

Worked example (no skipped steps)

14.9 Find 15% of 60.

Convert:

15% = 0 . 15

Multiply:

0 . 15 · 60 = 9

Likely skill: Chapter 4.

Worked example (no skipped steps)

14.10 Convert 0 . 08 to a percent.

Move the decimal point two places to the right:

0 . 08 = 8%

Likely skill: Chapter 4.

Worked example (no skipped steps)

14.11 Find the slope through (1 , −2) and (5 , 6).

Use

𝑦 2 − 𝑦 1

𝑚 = 𝑥 2 − 𝑥 1

So:

6 − (−2)

8

𝑚 =

=

= 2

5 − 1

4

Likely skill: Chapter 11.

Worked example (no skipped steps)

14.12 Solve 𝑦 = 𝑚𝑥 + 𝑏 for 𝑏.

Subtract 𝑚𝑥:

𝑏 = 𝑦 − 𝑚𝑥

Likely skill: Chapter 12.

Worked example (no skipped steps)

14.13 Solve 2( 𝑥 − 3) + 5 = 17.

Distribute:

2 𝑥 − 6 + 5 = 17

Combine:

2 𝑥 − 1 = 17

Add 1:

2 𝑥 = 18

198

Step-by-step solutions

Divide by 2:

𝑥 = 9

Likely skill: Chapter 7.

Worked example (no skipped steps)

14.14 Factor 𝑥 2 − 9.

This is a difference of squares:

𝑥 2 − 9 = 𝑥 2 − 32 = ( 𝑥 − 3)( 𝑥 + 3) Likely skill: Chapter 10.

Worked example (no skipped steps)

14.15 Solve 𝑦 + 6 = 9.

4

Subtract 6:

𝑦 = 3

4

Multiply by 4:

𝑦 = 12

Likely skill: Chapter 6.

Worked example (no skipped steps)

14.16 Find the slope through (−3 , 2) and (1 , −6).

−6 − 2

−8

𝑚 =

=

= −2

1 − (−3)

4

Likely skill: Chapter 11.

Worked example (no skipped steps)

14.17 Solve 𝐴 = 1 𝑏ℎ for 𝑏.

2

Multiply by 2:

2 𝐴 = 𝑏ℎ

Divide by ℎ:

2 𝐴

𝑏 =

( ℎ ̸= 0)

ℎ

Likely skill: Chapter 12.

Chapter 14

199

Worked example (no skipped steps)

14.18 Simplify 𝑥 2· 𝑥 3 .

𝑥

First combine the numerator:

𝑥 2 · 𝑥 3 = 𝑥 5

Now divide by 𝑥:

𝑥 5 = 𝑥 4

( 𝑥 ̸= 0)

𝑥

Likely skill: Chapter 9.

Worked example (no skipped steps)

14.19 Solve −3 𝑛 < 12.

Divide by −3 and flip the sign:

𝑛 > −4

Likely skill: Chapter 8.

Worked example (no skipped steps)

14.20 A number increased by 7 is 19. Find the number.

Let the number be 𝑥. Then:

𝑥 + 7 = 19

Subtract 7:

𝑥 = 12

Likely skill: Chapter 13.

Worked example (no skipped steps)

14.21 A $50 item gets an 8% sales tax. Find the total cost.

Multiply by:

1 . 08

So:

1 . 08 · 50 = 54

Answer: $54.

Likely skill: Chapter 4.

200

Step-by-step solutions

Worked example (no skipped steps)

14.22 Factor 2 𝑥 2 + 10 𝑥 + 12.

Take out the greatest common factor:

2 𝑥 2 + 10 𝑥 + 12 = 2( 𝑥 2 + 5 𝑥 + 6) Now factor the trinomial:

𝑥 2 + 5 𝑥 + 6 = ( 𝑥 + 2)( 𝑥 + 3) So:

2 𝑥 2 + 10 𝑥 + 12 = 2( 𝑥 + 2)( 𝑥 + 3) Likely skill: Chapter 10.

Worked example (no skipped steps)

14.23 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

Likely skill: Chapter 7.

Worked example (no skipped steps)

14.24 A rectangle has width 5 and perimeter 34. Find the length.

Let the length be 𝑙. Then:

34 = 2 𝑙 + 2(5)

Simplify:

34 = 2 𝑙 + 10

Subtract 10:

24 = 2 𝑙

Divide by 2:

𝑙 = 12

Likely skill: Chapter 13.

Chapter 14

201

Worked example (no skipped steps)

14.25 Find the slope through (2 , 5) and (6 , 13), then write the line with that slope and 𝑦-intercept −1.

First the slope:

13 − 5

8

𝑚 =

=

= 2

6 − 2

4

Now use slope-intercept form:

𝑦 = 𝑚𝑥 + 𝑏

With 𝑚 = 2 and 𝑏 = −1:

𝑦 = 2 𝑥 − 1

Likely skill: Chapter 11.

Worked example (no skipped steps)

14.26 Solve 𝐶 = 2 𝜋𝑟 for 𝑟, then find 𝑟 when 𝐶 = 10 𝜋.

First solve for 𝑟:

𝐶

𝑟 = 2 𝜋

Now substitute 𝐶 = 10 𝜋:

10 𝜋

𝑟 =

= 5

2 𝜋

Likely skill: Chapter 12.

Worked example (no skipped steps)

14.27 Solve 2 𝑥−1 = 5.

3

Multiply by 3:

2 𝑥 − 1 = 15

Add 1:

2 𝑥 = 16

Divide by 2:

𝑥 = 8

Likely skill: Chapter 7.

Worked example (no skipped steps)

14.28 Factor 4 𝑥 2 − 25.

Difference of squares:

4 𝑥 2 − 25 = (2 𝑥)2 − 52 = (2 𝑥 − 5)(2 𝑥 + 5) Likely skill: Chapter 10.

202

Step-by-step solutions

Worked example (no skipped steps)

14.29 A test score rises from 70 to 84. Find the percent increase.

Change:

84 − 70 = 14

Divide by the original:

14 = 0 . 2

70

Convert:

0 . 2 = 20%

Likely skill: Chapter 4.

Worked example (no skipped steps)

14.30 Two consecutive integers have sum 31. Find them.

Let the first be 𝑛. Then:

𝑛 + ( 𝑛 + 1) = 31

Combine:

2 𝑛 + 1 = 31

Subtract 1:

2 𝑛 = 30

Divide by 2:

𝑛 = 15

So the integers are 15 and 16.

Likely skill: Chapter 13.