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Grade 11 Precalculus · Lesson 1.1

Real Numbers · Sets, Intervals & Distance

Classify numbers, describe sets in different forms, operate on sets and intervals, and interpret absolute value as distance.
8 learning scenesInteractiveSame lesson material

Learn the ideas, see the mathematics, then practise the skill.

Key takeaway

Mr. Azzam’s lesson map

1 · Identify

What kind of number or set are you looking at?

2 · Represent

Translate between symbols, words, intervals, and number lines.

3 · Operate

Apply union, intersection, complement, or distance carefully.

Real Number System · classify from smallest family

A rational number can be written as a ratio of integers. Terminating and repeating decimals are rational. Nonterminating, nonrepeating decimals are irrational.

N⊂W⊂Z⊂Q⊂R\mathbb N\subset W\subset\mathbb Z\subset\mathbb Q\subset\mathbb R
ExplanationNatural & Whole

Natural numbers start at 1; whole numbers include 0.

ExplanationIntegers

Whole numbers and their negative opposites.

ExplanationRational

Can be written as a ratio of integers with nonzero denominator.

ExplanationRational−1.5,0,52,3.14‾,100-1.5,\quad 0,\quad \frac52,\quad 3.\overline{14},\quad 100
ExplanationIrrational−π,2,23-\pi,\quad \sqrt2,\quad \sqrt{23}
ExplanationClassification routine

Ask whether the number is natural, whole, integer, rational, or irrational. When the question asks for the most specific set, stop at the smallest family that contains it.

Try it · classify before revealing
−8,52,−π-8,\quad \frac{5}{2},\quad -\pi

Classify each number as specifically as possible.

Your turn · Integer

Work this step on paper first. The solution stays hidden until you confirm you are ready.

Your turn

Which number is irrational?
Common mistake. A decimal does not have to terminate to be rational. A repeating decimal is rational.

Sets & Set-Builder Notation · say exactly who belongs

Definition

Set

A set is a collection of objects; the objects are called elements of the set.
A={1,2,3,4,5,6}A=\{1,2,3,4,5,6\}
A={x∣x is an integer and 0<x<7}A=\{x\mid x\text{ is an integer and }0<x<7\}
ExplanationRoster → set-builder

Describe the type of elements first, then write the condition that selects exactly those elements. Set-builder → roster works in reverse: list every allowed element when the set is discrete and finite.

Complement · use the universal set
U={1,2,3,4,5,6,7,8,9},A={1,3,5,7,9}U=\{1,2,3,4,5,6,7,8,9\},\quad A=\{1,3,5,7,9\}

Find the elements in U that are not in A.

Your turn · Start with U

Work this step on paper first. The solution stays hidden until you confirm you are ready.

Your turn

Which roster form matches the set? B={x∈Z∣−2<x≤2}B=\{x\in\mathbb Z\mid -2<x\le2\}

Union & Intersection · OR versus BOTH

Union keeps everything in A or B or both. Intersection keeps only what the sets have in common.

ExplanationABU

A ∪ B · in A or B or both

ExplanationABU

A ∩ B · common to both

Sets · predict both results
S={1,2,3,4,5},T={4,5,6,7}S=\{1,2,3,4,5\},\quad T=\{4,5,6,7\}

Predict union and intersection first.

Your turn · Union = OR

Work this step on paper first. The solution stays hidden until you confirm you are ready.

Your turn

Find the intersection. A={1,3,5,7},B={3,4,5,8}A=\{1,3,5,7\},\quad B=\{3,4,5,8\}
Common mistake. Do not use the overlap for a union. Remember: union = OR; intersection = BOTH.
Interactive lab

Set + interval lab

Write each representation first, then operate. A continuous interval contains every real point between its endpoints, not just integers.

Elements of the set

A={−3,−1,0,2,4,6}A=\{-3,-1,0,2,4,6\}

Continuous interval

B=(−1,4]B=(-1,4]-5-4-3-2-1012345
Choose the intersection.

Intervals & Number Lines · four representations, one set

Inequality notation, interval notation, set-builder notation, and a number-line graph can all describe the same set.

ExplanationIncluded endpointa≤x⟺[a,…a\le x\quad\Longleftrightarrow\quad [a,\ldots

Closed dot · bracket.

ExplanationExcluded endpointa<x⟺(a,…a<x\quad\Longleftrightarrow\quad (a,\ldots

Open dot · parenthesis.

Interactive lab

Explore the interval

Move the endpoints, choose open or closed, and predict whether a test point belongs to the interval.
[−2,4)[-2,4)-5-4-3-2-1012345
Does -2 belong to this interval?
Interval intersection · keep the overlap
(1,3)∩[2,7](1,3)\cap[2,7]

Find the common part.

Your turn · Left boundary

Work this step on paper first. The solution stays hidden until you confirm you are ready.

Interval union · cover everything
(1,3)∪[2,7](1,3)\cup[2,7]

Keep every point covered by either interval.

Your turn · Overlap

Work this step on paper first. The solution stays hidden until you confirm you are ready.

Your turn

Convert the inequality to interval notation. −2<x≤1-2<x\le1
Common mistake. Infinity is never an included endpoint. Always use a parenthesis beside ∞\infty or −∞-\infty.

Absolute Value & Distance · decide the sign first

Definition

Absolute value

The absolute value of a real number is its distance from zero.
∣a∣={a,a≥0−a,a<0|a|=\begin{cases}a,&a\ge0\\-a,&a<0\end{cases}
d(a,b)=∣b−a∣=∣a−b∣d(a,b)=|b-a|=|a-b|
Explanation

Mr. Azzam’s comparison

∣a−b∣={b−a,a<ba−b,a>b|a-b|=\begin{cases}b-a,&a<b\\a-b,&a>b\end{cases}

Decide which quantity is larger first. Distance can never be negative.

Absolute value with a condition
a<b:∣a−b∣=?a<b:\quad |a-b|=?

Decide the sign inside first.

Your turn · Compare

Work this step on paper first. The solution stays hidden until you confirm you are ready.

Distance on the real line
d(−8,2)d(-8,2)

Try each checkpoint, or reveal all worked steps whenever you need them.

Your turn · Formula

Work this step on paper first. The solution stays hidden until you confirm you are ready.

Your turn

Given a<ba<b, simplify. a+b+∣a−b∣a+b+|a-b|
Common mistake. a−ba-b and b−ab-a are generally different, but ∣a−b∣=∣b−a∣|a-b|=|b-a|.

Exam Strategy · translate before you calculate

Key takeawaySTEP 1
Identify the representation

Set, roster, set-builder, interval, inequality, or number line?

Key takeawaySTEP 2
Translate carefully

Open/closed endpoints and integer restrictions change the set.

Key takeawaySTEP 3
Choose the operation

Union means OR; intersection means BOTH; complement uses U.

Key takeawaySTEP 4
Check the result

No repeated roster elements; distance nonnegative; infinity never closed.

Common mistakeMixed-representation warning

A finite set such as {−4,−2,0,1,3,5,7}\{-4,-2,0,1,3,5,7\} is not an interval. Test its individual elements against the interval before taking union or intersection.

Mastery Checkpoint · choose what needs practice

Rate each skill. Your choices seed Smart Practice, where the full lesson question bank continues with varied core and challenge questions.

Lesson Mastery Checkpoint

Rate each skill. This is only the starting signal; your actual Practice and Assessment performance will update the plan.

Classifying real numbers

Sets and set-builder notation

Union and intersection

Intervals and number-line notation

Absolute value and distance

Rate all skills to prepare your personalized practice path.
Key takeaway
Ready to move on?

You should be able to move between representations and explain why an endpoint is included or excluded—not only select an answer.

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