Algebra Foundations for Normal People by Gabriel Beguerie - HTML preview
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Chapter goals
By the end of this chapter, you will be able to:
• solve linear inequalities,
• know exactly when and why the inequality sign flips,
• read interval notation and number-line meaning,
• solve simple compound inequalities.
Prerequisites (if this is hard, go back to Unit 0)
You need equation-solving from Chapters 6–7 and signed-number order from Chapter 2.
Why this matters
Key idea
Equations ask for exact equality. Inequalities describe a whole region of values. That makes them useful for budgets, limits, safety ranges, admission cutoffs, and minimum requirements.
Real life: what it’s for
“At least 60% on the final.” “No more than 20 grams of sugar.” “Temperature below freezing.”
“Income greater than the rent requirement.”
Those are inequalities, not equations.
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Solve inequalities like equations — until a negative appears 69
Solve inequalities like equations — until a negative appears Recipe (steps that always work)
Solve a linear inequality almost the same way you solve an equation: 1. do the same legal move to both sides,
2. simplify carefully,
3. if you multiply or divide by a negative number, reverse the inequality sign, 4. read the answer as a region of values, not just a symbol string.
Worked example (no skipped steps)
Solve:
𝑥 + 4 > 9
Step 1: subtract 4 from both sides.
𝑥 + 4 − 4 > 9 − 4
Step 2: simplify both sides.
𝑥 > 5
Meaning: any number bigger than 5 works.
Worked example (no skipped steps)
Solve:
3 𝑥 − 5 ≥ 7
Step 1: add 5 to both sides.
3 𝑥 − 5 + 5 ≥ 7 + 5
3 𝑥 ≥ 12
Step 2: divide both sides by 3.
3 𝑥
12
≥
3
3
𝑥 ≥ 4
Meaning: 4 works, and every number larger than 4 also works.
Chapter 8. Inequalities without sign mistakes
Worked example (no skipped steps)
Solve:
−3 𝑥 ≤ 12
Step 1: divide both sides by −3. Normally division keeps the balance, but here the divisor is negative, so the inequality sign must reverse.
−3 𝑥
12
≥
−3
−3
Step 2: simplify.
𝑥 ≥ −4
Meaning: any number greater than or equal to −4 works.
Why the sign flips
Key idea
On the number line, multiplying by a negative reflects values across zero. That reflection reverses order.
For example, since
2 < 5 ,
multiplying by −1 gives
−2 > −5 .
That is why the sign must flip.
Note
This is not a rule to memorize blindly. It is a consequence of how negative multiplication changes direction on the number line. Once you see that, the flip stops feeling arbitrary.
Compound inequalities
Key idea
A compound inequality gives two conditions at once. For example, 1 < 𝑥 ≤ 5
means:
• 𝑥 is bigger than 1, and
• 𝑥 is at most 5.
Both parts must be true at the same time.
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Worked example (no skipped steps)
Solve:
2 < 𝑥 + 1 ≤ 6
Step 1: subtract 1 from all three parts.
2 − 1 < 𝑥 + 1 − 1 ≤ 6 − 1
Step 2: simplify.
1 < 𝑥 ≤ 5
Meaning: 𝑥 is bigger than 1 and at most 5.
Worked example (no skipped steps)
Solve:
−4 ≤ 2 𝑦 − 2 < 8
Step 1: add 2 to all three parts.
−4 + 2 ≤ 2 𝑦 − 2 + 2 < 8 + 2
−2 ≤ 2 𝑦 < 10
Step 2: divide all three parts by 2. Because 2 is positive, the inequality directions stay the same.
−1 ≤ 𝑦 < 5
Answer: −1 ≤ 𝑦 < 5.
Interval notation
Key idea
Interval notation is just a compact way to write a set of numbers.
• A parenthesis means the endpoint is not included.
• A bracket means the endpoint is included.
For example:
𝑥 > 3
becomes
(3 , ∞)
𝑥 ≥ 3
becomes
[3 , ∞)
Infinity is never an endpoint you can actually include, so ∞ and −∞ always use parentheses, never brackets.
Chapter 8. Inequalities without sign mistakes
Worked example (no skipped steps)
Write the solution
𝑥 ≥ −2
in interval notation.
Because −2 is included, use a bracket at −2. The values continue forever to the right, so use infinity:
[−2 , ∞)
Worked example (no skipped steps)
Write the solution
𝑥 < 4
in interval notation.
Because 4 is not included, use a parenthesis at 4. The values continue forever to the left, so: (−∞ , 4)
Common mistakes (and how to avoid them)
• forgetting to flip the sign when dividing by a negative,
• flipping the sign when dividing by a positive,
• solving the left and right parts of a compound inequality inconsistently,
• forgetting what the answer means on the number line,
• using a bracket when the endpoint is not included.
Recap: what you’re taking away
An inequality answer is usually not one number. It is a whole set of numbers.
So after solving, always ask:
• Does the direction of the sign make sense?
• Is the endpoint included or not?
• Can I describe the answer as a region on the number line?
Problems
8.1. Solve: 𝑥 + 4 > 9.
8.2. Solve: 𝑦 − 3 ≤ 7.
8.3. Solve: 2 𝑥 < 10.
8.4. Solve: −3 𝑥 ≤ 12.
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8.5. Solve: 5 − 2 𝑘 ≥ 1.
8.6. Solve: 2 < 𝑥 + 1 ≤ 6.
8.7. Write the solution to 𝑥 ≥ −2 in interval notation.
8.8. Write the solution to 𝑥 < 4 in interval notation.
8.9. Solve: 𝑥 + 4 > 11.
8.10. Solve: 3 𝑥 − 5 ≤ 7.
8.11. Solve: −2 𝑥 < 10.
8.12. Solve: 5 − 3 𝑥 ≥ −7.
8.13. Solve: −3 ≤ 𝑥 + 2 < 9.
8.14. Solve: −4 ≤ 2 𝑦 − 2 < 8.
8.15. Solve: 7 − 2 𝑚 > 1.
8.16. Solve: 𝑥 − 4 ≥ −1.
3
Quick checklist (if you miss it, reread and try again)
• I solve inequalities like equations until a negative multiplier or divisor appears.
• I know exactly when to reverse the sign.
• I can read a solution as a region, not just a symbol string.
Exponents and roots that actually
make sense