Algebra Foundations for Normal People by Gabriel Beguerie - HTML preview

Chapter goals

By the end of this chapter, you will be able to:

• identify terms, coefficients, and constants,

• combine like terms correctly,

• distribute across parentheses without dropping signs,

• simplify algebraic expressions in a repeatable way.

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

You need variables from Chapter 1 and signed-number control from Chapter 2.

Why this matters

Key idea

Expressions are the raw material of algebra. Before you solve equations or graph functions, you need to simplify what is on the page.

A lot of “harder math” is really expression control plus one new idea.

Real life: what it’s for

A phone plan might cost “base fee + price per GB.” A taxi model might be “start fee + price per mile.” A simple salary model might be “hourly rate times hours worked.”

Those are expressions. Simplifying expressions is what makes formulas usable instead of messy.

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Chapter 5. Expressions and the distributive property

Terms, coefficients, constants

Key idea

In

4 𝑥 − 7 + 3 𝑥 + 2

• the terms are 4 𝑥, −7, 3 𝑥, and 2,

• the coefficients are 4 and 3,

• the constants are −7 and 2.

Like terms can be combined. Unlike terms cannot.

Worked example (no skipped steps)

Simplify:

4 𝑥 − 7 + 3 𝑥 + 2

Step 1: group the like terms mentally. The 𝑥-terms go together and the constants go together.

Step 2: combine the 𝑥-terms.

4 𝑥 + 3 𝑥 = 7 𝑥

Step 3: combine the constants.

−7 + 2 = −5

Step 4: write the simplified expression.

4 𝑥 − 7 + 3 𝑥 + 2 = 7 𝑥 − 5

The distributive property

Recipe (steps that always work)

To distribute a number across parentheses, multiply it by every term inside: 𝑎( 𝑏 + 𝑐) = 𝑎𝑏 + 𝑎𝑐

The same idea works with subtraction:

𝑎( 𝑏 − 𝑐) = 𝑎𝑏 − 𝑎𝑐

The distributive property

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Worked example (no skipped steps)

Expand and simplify:

3( 𝑥 + 4)

Distribute the 3 to each term inside the parentheses:

3( 𝑥 + 4) = 3 · 𝑥 + 3 · 4

= 3 𝑥 + 12

Worked example (no skipped steps)

Expand and simplify:

−2(3 𝑥 − 5)

Step 1: distribute −2 to both terms.

−2(3 𝑥 − 5) = (−2)(3 𝑥) + (−2)(−5)

Step 2: multiply each part.

(−2)(3 𝑥) = −6 𝑥,

(−2)(−5) = 10

So the result is

−6 𝑥 + 10

Worked example (no skipped steps)

Expand and simplify:

2(3 𝑥 + 1) − 4

Step 1: distribute the 2.

2(3 𝑥 + 1) − 4 = 6 𝑥 + 2 − 4

Step 2: combine constants.

2 − 4 = −2

So the simplified expression is

6 𝑥 − 2

Worked example (no skipped steps)

Simplify:

7 − (2 𝑥 − 5)

Step 1: rewrite subtraction as adding the opposite.

7 − (2 𝑥 − 5) = 7 + [−1(2 𝑥 − 5)]

Step 2: distribute −1 to every term inside the parentheses.

−1(2 𝑥 − 5) = −2 𝑥 + 5

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Chapter 5. Expressions and the distributive property

Step 3: combine.

7 − (2 𝑥 − 5) = 7 − 2 𝑥 + 5 = 12 − 2 𝑥

Answer: 12 − 2 𝑥.

Note

A good habit is to circle or mentally mark like terms before combining them. It slows you down for five seconds and saves a lot of avoidable errors.

Bridge (why this matters later)

The distributive property appears everywhere later:

factoring, slope formulas, trig

simplifications, derivative rules, and statistics formulas all rely on clean expression handling.

Common mistakes (and how to avoid them)

• combining unlike terms,

• distributing to the first term but forgetting the second,

• losing a negative sign when distributing,

• simplifying in a random order instead of grouping like terms.

Recap: what you’re taking away

This chapter teaches two habits that pay off for the rest of the series:

• combine only what truly belongs together,

• when parentheses appear, touch every term inside.

If those habits become automatic, later algebra becomes much less fragile.

Problems

5.1. Simplify: 5 𝑥 + 2 𝑥.

5.2. Simplify: 7 𝑎 − 3 + 2 𝑎 + 8.

5.3. Expand: 4( 𝑥 + 3).

5.4. Expand: −3( 𝑦 − 2).

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

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

5.7. Decide whether 3 𝑥 and 3 are like terms.

5.8. Simplify: 6 𝑚 − 2 𝑚 + 9 − 4.

Problems

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5.9. Simplify: 4 𝑥 + 7 − 2 𝑥 + 3.

5.10. Expand: 3( 𝑥 − 5).

5.11. Simplify: −2(4 𝑦 − 1) + 3 𝑦.

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

5.13. Simplify: 3 𝑥 + 5 − 2 𝑥 + 7.

5.14. Distribute and simplify: 4( 𝑥 − 3).

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

5.16. Simplify: 2(3 𝑎 − 4) + 5 𝑎.

5.17. Simplify: 7 − (2 𝑥 − 5).

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

5.19. Simplify: 3(2 𝑝 + 5) − 4( 𝑝 − 1).

5.20. Simplify: 5 − 3( 𝑥 − 2).

Quick checklist (if you miss it, reread and try again)

• I know what a term is.

• I only combine like terms.

• I can distribute across parentheses without dropping signs.

Chapter 6

Solving one-step and two-step

equations