Algebra Foundations for Normal People by Gabriel Beguerie - HTML preview

Chapter 15

Worked example (no skipped steps)

15.1 Simplify 4 𝑥 − 3 + 2 𝑥 + 9.

4 𝑥 + 2 𝑥 = 6 𝑥,

−3 + 9 = 6

So:

6 𝑥 + 6

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Worked example (no skipped steps)

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

Distribute:

−6 𝑥 + 8 + 5

Combine:

−6 𝑥 + 13

Worked example (no skipped steps)

15.3 Solve 3 𝑥 + 7 = 22.

Subtract 7:

3 𝑥 = 15

Divide by 3:

𝑥 = 5

Worked example (no skipped steps)

15.4 Solve 5( 𝑥 − 1) = 20.

Distribute:

5 𝑥 − 5 = 20

Add 5:

5 𝑥 = 25

Divide by 5:

𝑥 = 5

Worked example (no skipped steps)

15.5 Solve −4 𝑥 ≥ 12.

Divide by −4 and flip the sign:

𝑥 ≤ −3

Worked example (no skipped steps)

15.6 Simplify 3 + 1 .

4

8

3

6

=

4

8

So:

6

1

7

+

=

8

8

8

204

Step-by-step solutions

Worked example (no skipped steps)

15.7 Simplify 𝑎 3 · 𝑎 4.

𝑎 3+4 = 𝑎 7

Worked example (no skipped steps)

15.8 Factor 𝑥 2 + 8 𝑥 + 15.

Numbers that multiply to 15 and add to 8 are 3 and 5:

𝑥 2 + 8 𝑥 + 15 = ( 𝑥 + 3)( 𝑥 + 5) Worked example (no skipped steps)

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

6 − (−2)

8

𝑚 =

=

= 2

5 − 1

4

Worked example (no skipped steps)

15.10 Solve 𝑦 = 𝑚𝑥 + 𝑏 for 𝑏.

Subtract 𝑚𝑥:

𝑏 = 𝑦 − 𝑚𝑥

Worked example (no skipped steps)

15.11 Find 12% of 150.

12% = 0 . 12

Then:

0 . 12 · 150 = 18

Answer: 18.

Worked example (no skipped steps)

15.12 Convert 0 . 375 to a percent.

Move the decimal point two places to the right:

0 . 375 = 37 . 5%

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Worked example (no skipped steps)

15.13 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)

15.14 Factor 4 𝑥 2 − 25.

Difference of squares:

4 𝑥 2 − 25 = (2 𝑥)2 − 52 = (2 𝑥 − 5)(2 𝑥 + 5) Worked example (no skipped steps)

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

Let the number be 𝑥. Then:

𝑥 + 7 = 19

Subtract 7:

𝑥 = 12

Worked example (no skipped steps)

15.16 A rectangle has length 8 and perimeter 26. Find the width.

Let the width be 𝑤. Then:

26 = 2(8) + 2 𝑤

So:

26 = 16 + 2 𝑤

Subtract 16:

10 = 2 𝑤

Divide by 2:

𝑤 = 5

206

Step-by-step solutions

Worked example (no skipped steps)

15.17 Simplify 𝑥 2· 𝑥 3 .

𝑥

𝑥 2 · 𝑥 3 = 𝑥 5

Now divide by 𝑥:

𝑥 5 = 𝑥 4

( 𝑥 ̸= 0)

𝑥

Worked example (no skipped steps)

15.18 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)

15.19 Solve −3 𝑛 < 12.

Divide by −3 and flip the sign:

𝑛 > −4

Worked example (no skipped steps)

15.20 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:

𝑥 2 + 5 𝑥 + 6 = ( 𝑥 + 2)( 𝑥 + 3) So:

2 𝑥 2 + 10 𝑥 + 12 = 2( 𝑥 + 2)( 𝑥 + 3)

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Worked example (no skipped steps)

15.21 Find the slope through (2 , 5) and (6 , 13).

13 − 5

8

𝑚 =

=

= 2

6 − 2

4

Worked example (no skipped steps)

15.22 Solve 𝐴 = 1 𝑏ℎ for 𝑏.

2

Multiply by 2:

2 𝐴 = 𝑏ℎ

Divide by ℎ:

2 𝐴

𝑏 =

( ℎ ̸= 0)

ℎ

Worked example (no skipped steps)

15.23 A $40 shirt is discounted by 30%. Find the sale price.

A 30% discount leaves 70%:

0 . 70 · 40 = 28

Answer: $28.

Worked example (no skipped steps)

15.24 A 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%

Answer: 20%.

Worked example (no skipped steps)

15.25 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

208

Step-by-step solutions

Subtract 2:

𝑥 = 7

Worked example (no skipped steps)

15.26 Solve 𝑥 + 𝑥 = 5.

2

3

Multiply every term by 6:

3 𝑥 + 2 𝑥 = 30

So:

5 𝑥 = 30

Divide by 5:

𝑥 = 6

Worked example (no skipped steps)

15.27 Decide whether 2 𝑥 + 5 = 2 𝑥 − 1 has one solution, no solution, or infinitely many solutions.

Subtract 2 𝑥 from both sides:

5 = −1

This is false, so there is no solution.

Worked example (no skipped steps)

15.28 Solve for 𝐶 in 𝐹 = 9 𝐶 + 32.

5

Subtract 32:

9

𝐹 − 32 =

𝐶

5

Multiply by 5 :

9

5

𝐶 =

( 𝐹 − 32)

9

Worked example (no skipped steps)

15.29 Two consecutive integers have sum 31. Find them.

Let the first integer be 𝑛. Then:

𝑛 + ( 𝑛 + 1) = 31

So:

2 𝑛 + 1 = 31

Subtract 1:

2 𝑛 = 30

Divide by 2:

𝑛 = 15

The integers are 15 and 16.

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Worked example (no skipped steps)

15.30 A rope 36 meters long is cut into two pieces. One is 8 meters longer than the other.

Find both lengths.

Let the shorter piece be 𝑥. Then:

𝑥 + ( 𝑥 + 8) = 36

So:

2 𝑥 + 8 = 36

Subtract 8:

2 𝑥 = 28

Divide by 2:

𝑥 = 14

The longer piece is 22, so the two pieces are 14 and 22.

Worked example (no skipped steps)

15.31 Solve 5 − 2( 𝑥 + 1) = −9.

Distribute:

5 − 2 𝑥 − 2 = −9

Combine:

3 − 2 𝑥 = −9

Subtract 3:

−2 𝑥 = −12

Divide by −2:

𝑥 = 6

Worked example (no skipped steps)

15.32 Explain in one sentence why a vertical line has undefined slope.

Slope is

change in 𝑦 .

change in 𝑥

On a vertical line, the change in 𝑥 is 0, and division by 0 is undefined, so the slope is undefined.