Algebra Foundations for Normal People by Gabriel Beguerie - HTML preview
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Chapter goals
This chapter checks whether the algebra floor now feels stable enough for the next book. You should be able to handle:
• expressions and simplification,
• equations and inequalities,
• fractions, exponents, and factoring,
• slope and straight-line basics,
• formulas and basic translation from words to algebra.
Prerequisites (if this is hard, go back to Unit 0)
You need the full book. This chapter is an exit check, not a first lesson.
Key idea
You do not need perfection before opening the next book. You do need this: the basic moves should now feel repeatable instead of mysterious.
Real life: what it’s for
The next book asks you to think about functions, graphs, domain, range, transformations, and behavior. All of that becomes easier when the algebra underneath it no longer steals your attention.
Bridge (why this matters later)
Readiness is not “I never make mistakes.”
Readiness is “I know what move to try, I can usually do it, and when I miss one, I can locate 120
121
the weak spot.” That is the kind of readiness that lets the rest of the series feel challenging without feeling chaotic.
Minimum-ready check
Try these without looking back unless you truly freeze.
15.1. Simplify: 4 𝑥 − 3 + 2 𝑥 + 9.
15.2. Expand and simplify: −2(3 𝑥 − 4) + 5.
15.3. Solve: 3 𝑥 + 7 = 22.
15.4. Solve: 5( 𝑥 − 1) = 20.
15.5. Solve: −4 𝑥 ≥ 12.
15.6. Simplify: 3 + 1 .
4
8
15.7. Simplify: 𝑎 3 · 𝑎 4.
15.8. Factor: 𝑥 2 + 8 𝑥 + 15.
15.9. Find the slope through (1 , −2) and (5 , 6).
15.10. Solve 𝑦 = 𝑚𝑥 + 𝑏 for 𝑏.
15.11. Find 12% of 150.
15.12. Convert 0 . 375 to a percent.
15.13. Solve: 2 𝑥−1 = 5.
3
15.14. Factor: 4 𝑥 2 − 25.
15.15. A number increased by 7 is 19. Find the number.
15.16. A rectangle has length 8 and perimeter 26. Find the width.
Strong-ready check
If you want the next book to feel smoother, do this second set too.
15.17. Simplify: 𝑥 2· 𝑥 3 .
𝑥
15.18. Solve: 4 − 3(2 𝑥 − 1) = −11.
15.19. Solve: −3 𝑛 < 12.
15.20. Factor: 2 𝑥 2 + 10 𝑥 + 12.
15.21. Find the slope through (2 , 5) and (6 , 13).
15.22. Solve 𝐴 = 1 𝑏ℎ for 𝑏.
2
15.23. A $40 shirt is discounted by 30%. Find the sale price.
Chapter 15. Readiness check for Math Functions
15.24. A score rises from 70 to 84. Find the percent increase.
15.25. Solve: 2( 𝑥 + 3) − 4 = 𝑥 + 9.
15.26. Solve: 𝑥 + 𝑥 = 5.
2
3
15.27. Decide whether 2 𝑥 + 5 = 2 𝑥 − 1 has one solution, no solution, or infinitely many solutions.
15.28. Solve for 𝐶 in 𝐹 = 9 𝐶 + 32.
5
15.29. Two consecutive integers have sum 31. Find them.
15.30. A rope 36 meters long is cut into two pieces. One is 8 meters longer than the other. Find both lengths.
15.31. Solve: 5 − 2( 𝑥 + 1) = −9.
15.32. Explain in one sentence why a vertical line has undefined slope.
How to score it
What to report (so your result is useful)
Use the two parts honestly.
Minimum-ready check only
• 14–16 correct: ready to begin Math Functions for Normal People.
• 11–13 correct: almost ready; repair the misses first.
• 10 or fewer correct: do more repair before moving on.
Whole chapter (all 32 problems)
• 28–32 correct: strong ready.
• 23–27 correct: ready, but review the misses first.
• 18–22 correct: shaky ready; repair the weak spots before moving on.
• Below 18 correct: stay here longer. The problem is not talent. The floor still needs reinforcement.
Red-flag misses
Note
Some misses matter more than others.
If you miss several of these, repair them before moving on:
• fractions and percent translation,
• distribution with negatives,
• equations with variables on both sides,
Where to repair if you miss certain problems 123
• factoring,
• slope,
• formula rearranging,
• word-problem translation.
Those are the moves that most strongly protect the next book.
Where to repair if you miss certain problems
Note
Use misses as information:
• Problems 1 and 2: review Chapter 5.
• Problems 3, 4, 13, 18, 25, 26, and 31: review Chapters 6 and 7.
• Problems 5 and 19: review Chapter 8.
• Problems 6, 11, 12, 23, and 24: review Chapters 3 and 4.
• Problems 7 and 17: review Chapter 9.
• Problems 8, 14, and 20: review Chapter 10.
• Problems 9, 21, and 32: review Chapter 11.
• Problems 10, 22, and 28: review Chapter 12.
• Problems 15, 16, 29, and 30: review Chapter 13.
Recap: what you’re taking away
Readiness means repeatability, not perfection.
If you can do most of this chapter without panic, and if you can locate your weak spot when you miss one, you are where you need to be. That is enough to start the next book well.
Quick checklist (if you miss it, reread and try again)
• I can do most of these without panic.
• When I miss a problem, I can identify which earlier chapter to review.
• I understand that readiness means repeatability, not perfection.
Step-by-step solutions (no skipped
steps)
Key idea
This chapter includes solutions to all end-of-chapter exercises in the book.
Use it well:
• first try the problem on your own,
• then compare your steps,
• finally ask: where did my logic diverge?
Looking at a solution is useful only if it changes what you can do next time.
Unit 0
Worked example (no skipped steps)
0.1 Evaluate 20 − 4 · 3.
Multiply first:
4 · 3 = 12
Then subtract:
20 − 12 = 8
Answer: 8.
Worked example (no skipped steps)
0.2 Evaluate 24 ÷ 6 · 2.
Division and multiplication share a level, so go left to right: 24 ÷ 6 = 4
4 · 2 = 8
Answer: 8.
124
125
Worked example (no skipped steps)
0.3 Evaluate (18 − 6) ÷ 3.
Parentheses first:
18 − 6 = 12
Then divide:
12 ÷ 3 = 4
Answer: 4.
Worked example (no skipped steps)
0.4 Evaluate 7 + 3 · 4.
Multiply first:
3 · 4 = 12
Then add:
7 + 12 = 19
Answer: 19.
Worked example (no skipped steps)
0.5 Decide whether 8 + 5 = 10 + 3 is true.
Left side:
8 + 5 = 13
Right side:
10 + 3 = 13
Both sides are equal, so the statement is true.
Worked example (no skipped steps)
0.6 Decide whether 9 − 2 = 3 + 5 is true.
Left side:
9 − 2 = 7
Right side:
3 + 5 = 8
Since 7 ̸= 8, the statement is false.
Step-by-step solutions
Worked example (no skipped steps)
0.7 Compute 5 − 9.
Rewrite subtraction as addition of the opposite:
5 − 9 = 5 + (−9)
Now combine:
5 + (−9) = −4
Answer: −4.
Worked example (no skipped steps)
0.8 Compute −8 − 5.
Rewrite as addition of the opposite:
−8 − 5 = −8 + (−5)
Now add two negative numbers:
−8 + (−5) = −13
Answer: −13.
Worked example (no skipped steps)
0.9 Compute −7 + 12.
Start at −7. Adding 12 means move 12 units to the right. That lands at: 5
Answer: 5.
Worked example (no skipped steps)
0.10 Compute (−6)(+2).
A negative times a positive is negative. Then multiply the absolute values: 6 · 2 = 12
So:
(−6)(+2) = −12
Answer: −12.
127
Worked example (no skipped steps)
0.11 Compute (−3)(−8).
Two negatives multiplied give a positive result. Then multiply the absolute values: 3 · 8 = 24
So:
(−3)(−8) = 24
Answer: 24.
Worked example (no skipped steps)
0.12 Explain in words what the equal sign means.
The equal sign means that the left side and the right side have the same value. It means balance, not “the answer starts here.”
Worked example (no skipped steps)
0.13 Write 3 as a division statement.
5
A fraction is a division. So:
3 = 3 ÷ 5
5
Worked example (no skipped steps)
0.14 Write 1 as a fraction with denominator 4.
We want a fraction equal to 1 with denominator 4. Since: 4 = 4 ÷ 4 = 1
4
we can write:
4
1 = 4
Worked example (no skipped steps)
0.15 Rewrite 1 − 1 so both numbers are written in fourths.
4
Write 1 as 4 :
4
4
1 = 4
Now substitute:
1
4
1
1 −
=
−
4
4
4
Step-by-step solutions
Worked example (no skipped steps)
0.16 Explain why 7 is greater than 1.
2
Because:
7 = 7 ÷ 2
2
and 7 ÷ 2 is bigger than 1. Another way to see it is that: 2 = 1
2
so seven halves are more than one whole.
Worked example (no skipped steps)
0.17 Rewrite 7 as a mixed number.
2
Break the seven halves into three full groups of 2 plus one half: 2
7
2
2
2
1
=
+
+
+
2
2
2
2
2
Now replace each 2 with 1:
2
7
1
= 1 + 1 + 1 +
2
2
Add the whole numbers:
1 + 1 + 1 = 3
So:
7
1
1
= 3 +
= 3
2
2
2
Worked example (no skipped steps)
0.18 Rewrite 1 as a fraction with denominator 10.
2
To change the denominator from 2 to 10, multiply by 5:
2 · 5 = 10
So multiply the numerator by the same factor:
1
1 · 5
5
=
=
2
2 · 5
10
129
Worked example (no skipped steps)
0.19 Rewrite 3 as a fraction with denominator 12.
4
To change the denominator from 4 to 12, multiply by 3:
4 · 3 = 12
So:
3
3 · 3
9
=
=
4
4 · 3
12
Worked example (no skipped steps)
0.20 Simplify 30 .
60
Both numbers are divisible by 30. Divide top and bottom by 30: 30
30 ÷ 30
1
=
=
60
60 ÷ 30
2
Answer: 1 .
2
Worked example (no skipped steps)
0.21 Simplify 12 .
18
Both numbers are divisible by 6. Divide top and bottom by 6: 12
12 ÷ 6
2
=
=
18
18 ÷ 6
3
Answer: 2 .
3
Worked example (no skipped steps)
0.22 Compute 2 + 5 .
9
9
The denominators already match, so add the numerators:
2
5
7
+
=
9
9
9
Answer: 7 .
9
Step-by-step solutions
Worked example (no skipped steps)
0.23 Compute 1 + 1 .
3
6
Use denominator 6. Rewrite 1 in sixths:
3
1
2
=
3
6
Now add:
2
1
3
+
=
6
6
6
Simplify by dividing top and bottom by 3:
3
3 ÷ 3
1
=
=
6
6 ÷ 3
2
Answer: 1 .
2
Worked example (no skipped steps)
0.24 Compute 1 + 3 .
4
8
The denominators do not match. A common denominator is 8 because 4 can be turned into 8
by multiplying by 2, while the second fraction already has denominator 8. Rewrite the first fraction:
1
2
=
4
8
The second fraction stays the same:
3
3
=
8
8
Now add:
2
3
5
+
=
8
8
8
Answer: 5 .
8
Worked example (no skipped steps)
0.25 Compute 1 − 1 .
4
Write 1 as fourths:
4
1 = 4
Now subtract:
4
1
3
−
=
4
4
4
Answer: 3 .
4
131
Worked example (no skipped steps)
0.26 Compute 5 − 1 .
6
4
A common denominator is 12 because both denominators divide 12. Rewrite both fractions: 5
10
1
3
=
=
6
12
4
12
Now subtract:
10
3
7
−
=
12
12
12
Answer: 7 .
12
Worked example (no skipped steps)
0.27 Compute 3 · 10 .
5
12
Multiply numerators and denominators:
3 · 10
30
=
5 · 12
60
Divide the numerator and denominator by 30 because 30 is a common factor of both: 30
30 ÷ 30
1
=
=
60
60 ÷ 30
2
Answer: 1 .
2
Worked example (no skipped steps)
0.28 Compute 7 ÷ 1 .
8
4
Change division by a fraction into multiplication by its reciprocal: 7
1
7
4
÷
=
·
8
4
8
1
Now multiply:
7 · 4
28
=
8 · 1
8
Simplify by dividing top and bottom by 4:
28
28 ÷ 4
7
=
=
8
8 ÷ 4
2
Use the mixed-number idea from Unit 0:
7
1
= 3
2
2
Answer: 7 or 3 1 .
2
2
Step-by-step solutions
Worked example (no skipped steps)
0.29 Convert 0 . 45 to a percent.
Multiply by 100:
0 . 45 · 100 = 45
So:
0 . 45 = 45%
Worked example (no skipped steps)
0.30 Convert 0 . 125 to a percent.
Multiply by 100:
0 . 125 · 100 = 12 . 5
So:
0 . 125 = 12 . 5%
Worked example (no skipped steps)
0.31 Convert 18% to a decimal.
Write the percent over 100:
18
18% = 100
Now divide:
18 = 0 . 18
100
Worked example (no skipped steps)
0.32 Convert 35% to a decimal.
Write the percent over 100:
35
35% =
= 0 . 35
100
Worked example (no skipped steps)
0.33 Find 25% of 60.
Convert the percent:
25% = 0 . 25
Now multiply because “of” means multiply:
0 . 25 · 60 = 15
Answer: 15.
133
Worked example (no skipped steps)
0.34 Find 12% of 250.
Convert the percent:
12% = 0 . 12
Now multiply:
0 . 12 · 250 = 30
Answer: 30.