Algebra Foundations for Normal People by Gabriel Beguerie - HTML preview

Chapter goals

This chapter helps you practice the real exam skill:

• choose the tool,

• spot the likely mistake point,

• check the result before moving on.

Prerequisites (if this is hard, go back to Unit 0)

You need the full toolkit from Chapters 0–13. This chapter mixes the skills on purpose.

Key idea

Real problems do not arrive labeled “this is a fractions problem” or “this is a factoring problem.”

Part of getting better at algebra is learning to choose the tool instead of waiting for the page to choose it for you.

Real life: what it’s for

This is the closest chapter to what quizzes, exams, placement tests, and later books usually feel like: mixed tasks, changing formats, and no giant arrow telling you which chapter the idea came from.

Recipe (steps that always work)

Before touching the pencil, ask:

1. Is this simplification, solving, factoring, substitution, or interpretation?

2. Are there fractions, signs, exponents, or parentheses that create risk?

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Chapter 14. Mixed review

3. What would a reasonable answer look like before I compute?

4. How can I check the result quickly at the end?

Bridge (why this matters later)

Mixed review is where the next book begins.

In Math Functions for Normal People, the page will stop telling you exactly which earlier chapter the skill came from. You will read a problem, choose a tool, and move calmly. This chapter is practice for that.

Note

If this chapter feels harder than the earlier ones, that does not mean you got worse. It means the training wheels are coming off. That is normal.

Mixed warm-up

These are shorter problems meant to wake the tools back up.

14.1. Simplify: 4 𝑥 − 3 + 2 𝑥 + 9.

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

14.3. Solve: 3 𝑥 + 7 = 22.

14.4. Solve: 5( 𝑥 − 1) = 20.

14.5. Solve: −4 𝑥 ≥ 12.

14.6. Simplify: 3 + 1 .

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8

14.7. Simplify: 𝑎 3 · 𝑎 4.

14.8. Factor: 𝑥 2 + 8 𝑥 + 15.

14.9. Find 15% of 60.

14.10. Convert 0 . 08 to a percent.

Mixed standard set

These begin to mix topics in a more exam-like way.

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

14.12. Solve 𝑦 = 𝑚𝑥 + 𝑏 for 𝑏.

14.13. Solve: 2( 𝑥 − 3) + 5 = 17.

14.14. Factor: 𝑥 2 − 9.

14.15. Solve: 𝑦 + 6 = 9.

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Mastery set

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

14.17. Solve 𝐴 = 1 𝑏ℎ for 𝑏.

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14.18. Simplify: 𝑥 2· 𝑥 3 .

𝑥

14.19. Solve: −3 𝑛 < 12.

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

Mastery set

These are meant to feel like the kind of mixed pressure that shows up before the next book.

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

14.22. Factor: 2 𝑥 2 + 10 𝑥 + 12.

14.23. Solve: 4 − 3(2 𝑥 − 1) = −11.

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

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

−1.

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

14.27. Solve: 2 𝑥−1 = 5.

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14.28. Factor: 4 𝑥 2 − 25.

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

14.30. Two consecutive integers have sum 31. Find them.

Common mistakes (and how to avoid them)

• doing the right chapter rule in the wrong place,

• not noticing a negative sign or denominator,

• forgetting whether the task asks for simplification or solving,

• rushing because the problem looks familiar.

Recap: what you’re taking away

If this chapter feels rough, that is information, not failure.

The weak spots you find here are the exact spots to repair before the next book. Mixed review is not punishment. It is diagnosis.

Chapter 15

Readiness check for Math Functions

for Normal People