Algebra · Lesson 5

Inequalities: claims about regions

An inequality is a claim about an ordered region, not a single number. Multiplying by a negative flips the order; intervals name the solution set.

Core ~15 minDeep study ~43 minWhy does multiplying by a negative flip an inequality?What is an interval as a solution set?

Before this: Equations as relationships, Graphs: seeing relationships

How this idea was born

Equality answers "are these the same?" Life often asks a different question: "is this enough?", "is this too fast?", "is this under the limit?" Budgets, bridges, medicine doses, and exam cutoffs care about regions of allowed values, not a single perfect number.

Ancient and medieval mathematicians compared magnitudes constantly (larger, smaller, between), but modern compact symbols took centuries. The symbols < and > first appear in print in Thomas Harriot's Artis Analyticae Praxis (1631), published after his death. MacTutor and the Miller symbols pages urge care: Harriot's surviving manuscripts used different signs, and editors assembled the Praxis. Do not flatten that into "Harriot invented inequality." Steal the ethic instead: make order claims as crisp as equality claims.

For a modern rereader, an inequality is a truth claim about order. Solving it means naming every value that keeps the claim true - usually an interval or a union of intervals on the number line.

First principles

An inequality asserts an order relation between expressions, such as

F(x) > G(x), \quad F(x) \le G(x)

Solving means finding the solution set: every allowed

x

for which the claim holds.

Preserving versus flipping. Adding the same quantity to both sides preserves order. Multiplying or dividing both sides by a positive number preserves order. Multiplying or dividing by a negative number reverses the inequality direction.

Why? On the number line, multiplying by

-1

reflects through the origin. Reflection reverses left and right, so "left of" becomes "right of."

Intervals name solution sets compactly:

| Notation | Meaning | |----------|---------| |

(a, b)

a < x < b

| |

[a, b]

a \le x \le b

| |

(a, \infty)

x > a

| |

(-\infty, b]

x \le b

Open ends exclude the endpoint; closed ends include it. Infinity is never a number you "reach," only a way to say unbounded.

Integers add direction: left of zero is debt / below / opposite.

Worked intuition

Linear inequalities

Solve

3x - 2 < 7

. Add 2:

3x < 9

. Divide by 3 (positive):

x < 3

. Solution:

(-\infty, 3)

Now solve

-2x + 1 \ge 5

. Subtract 1:

-2x \ge 4

. Divide by

-2

and flip:

x \le -2

. Solution:

(-\infty, -2]

Check: pick

x = -3

(should work) and

x = 0

(should fail) in the second inequality.

Compound claims and unions

1 < x \le 4

is the half-open interval

(1, 4]

. "

x < 0

x > 2

" is

(-\infty, 0) \cup (2, \infty)

Region on a graph

For one variable, shade the number line. For two variables,

y > 2x + 1

shades a half-plane above the line

y = 2x + 1

(dashed if strict). Graphs from the previous chapter become boundaries of regions.

Quadratic preview

x^2 - 1 > 0

factors as

(x-1)(x+1) > 0

. The product is positive outside the roots:

x < -1

x > 1

. Algebrica's quadratic inequalities page walks sign charts carefully; use them when parabolas return in the next lesson.

Common confusions

Forgetting to flip when multiplying by a negative. The number-one trap. Reflect on the number line until the flip feels inevitable.

Treating inequalities like equations with one answer. Equations often isolate a point. Inequalities isolate a set.

Mixing open and closed endpoints. If the original claim is strict (<), do not include the boundary unless a separate argument justifies it.

Dividing by an expression that might be negative. If you divide by

(x-2)

, you must case-split on the sign of

(x-2)

, or rearrange differently.

Writing $\infty$ as if it were a real number you can plug in. Interval notation uses

\infty

as a direction marker, not a value.

History and stories

People compared lengths, weights, and times long before algebra textbooks. Greek geometry spoke in "greater than" language without modern symbols. What changed in early modern Europe was compact symbolic algebra: Recorde's = (1557), then printed order signs in the Harriot circle's Praxis (1631).

MacTutor's Harriot biography stresses two honest points at once: Harriot's equation-solving was remarkably advanced, and the attribution of < and > is historically delicate because the printed book was edited after his death. Prefer "first appear in the 1631 Praxis" over "Harriot invented <."

The deeper story is conceptual: once equations were claims you could transform, inequalities became claims you could transform, with one extra rule about orientation.

Real world

Speed limits: $v \le 60$ mph is a closed half-line of legal speeds.
Medication windows: concentration must satisfy $a \le c \le b$ .
Budgets: spending $s$ with $s \le B$ .
Tolerances in manufacturing: $|x - x_0| \le \varepsilon$ is a closed interval around the target.
Grading cutoffs: score $\ge 70$ to pass.

Every threshold policy is an inequality wearing a necktie.

Open when you want the machine-learning connection. Skip freely.

Field notes, extra examples, and glossary live here when you want more than the core path.

Carry this forward

Inequalities claim order; solutions are regions (often intervals).
Multiplying by a negative flips the claim.
Graphs turn inequalities into half-planes and shaded sets.
Next: quadratics, where parabola geometry meets polynomial algebra.

Go deeper when curiosity hits - videos, essays, and primary trails.

referenceMacTutor - Thomas HarriotEquations work and careful notes on who introduced < and > in print.
referenceMacTutor - Earliest Uses of Symbols of RelationHistory of =, <, > and the Praxis attribution debate.
articleAlgebrica - InequalitiesOrder relations, equivalence, and solution sets.
articleAlgebrica - Quadratic inequalitiesSign analysis and the flip when leading coefficient is negative.
bookSteven Strogatz - The Joy of XOrder and comparison as everyday mathematical muscle.

Lesson threads

Real world

Budgets, speed limits, dosing ranges, and pass/fail cutoffs are inequalities: allowed regions, not single answers.

Mathematician

Thomas Harriot

1560-1621

Harriot solved equations with a clarity that feels modern: completing the square, respecting both roots, treating negatives seriously. The famous < and > symbols appear in his posthumous Praxis (1631), though MacTutor notes the manuscript evidence is messy - Harriot himself used different signs, and editors shaped the printed book. The ethic worth stealing is careful notation in service of clear claims, not a slogan that one person alone invented inequality.

Later bridge · AI / ML

Optional trail - open when useful.

Constraints in optimization (weights within bounds, probabilities between 0 and 1, logits clipped) are inequality claims. Feasible regions are geometry.

Inequalities: claims about regions

How this idea was born

First principles

Worked intuition

Linear inequalities

Compound claims and unions

Region on a graph

Quadratic preview

Common confusions

History and stories

Real world

Carry this forward

Read more

Check your understanding