Grade 11 Precalculus · Chapter 1

Chapter 1 Review · Fundamentals

Review the rules, choose the method yourself, then finish with mixed mastery practice across Lessons 1.1, 1.3, 1.4, 1.8, and 1.9.

How to use this review

1 · Recall

Read the key rule without looking at a worked solution.

2 · Choose

Decide which method or pattern fits the structure.

3 · Prove mastery

Complete the mixed practice without a lesson label telling you the method.

1.1 · Real Numbers, Sets & Intervals

Union: in A or B or both. Intersection: common to both. Translate freely between inequality notation, interval notation, set notation, and a number line.

\mathbb N\subset W\subset\mathbb Z\subset\mathbb Q\subset\mathbb R

d(a,b)=|b-a|=|a-b|

Mr. Azzam reminder: if

a<b

, then

|a-b|=b-a

. Distance cannot be negative.

Quick Check

Check the idea before moving on.

Keep only the common overlap.

(-3,4]\cap[1,7)

1.3 · Algebraic Expressions & Factoring

Read the structure before doing algebra: exponent rules → operations → special products → factoring. For factoring, use the classroom routine rather than guessing.

Mr. Azzam’s factoring system

Check the GCF first.
Count the terms.
Choose the matching pattern.
Factor.
Ask: “Can I factor again?”

(A-B)(A+B)=A^2-B^2

(A\pm B)^2=A^2\pm2AB+B^2

A^3-B^3=(A-B)(A^2+AB+B^2)

A^3+B^3=(A+B)(A^2-AB+B^2)

Repeated complete expressions may be treated as one object using substitution. Rational and negative rational exponents are part of the chapter skill set.

Quick Check

Check the idea before moving on.

What is the FIRST question you ask before choosing a factoring pattern?

12x^3-48x

1.4 · Rational Expressions

Restrictions come from the original denominator. Factor before cancelling, and cancel factors — never terms joined by addition or subtraction.

Operations routine: Multiply → factor and cancel. Divide → Keep–Change–Flip, then factor. Add/Subtract → factor denominators, build the LCD, rewrite, combine, simplify.

Mr. Azzam’s preferred compound-fraction method: find one LCD for all small denominators and multiply the entire numerator and entire denominator by it.

a^{-m}=\frac1{a^m}

For a two-term radical numerator or denominator, use its conjugate when rationalizing.

Quick Check

Check the idea before moving on.

What must be recorded BEFORE cancellation?

\frac{x^2-9}{x^2+x-6}

1.8 · Inequalities

For linear inequalities, remember that multiplying or dividing by a negative reverses the sign. For nonlinear and rational inequalities, use critical numbers and a sign chart.

Mr. Azzam’s sign-chart routine

Move everything to one side and compare with 0.
Factor completely.
Find critical numbers.
Split the number line into intervals.
Determine signs and choose the required intervals.
Check endpoints.

Rational inequalities: ZN may be included when equality is allowed; ZD is never included. Do not cross-multiply by an expression of unknown sign.

|u|<c\iff-c<u<c\qquad |u|>c\iff u>c \text{or} u<-c

Quick Check

Check the idea before moving on.

Which critical value can NEVER be included?

\frac{(x-2)(x+1)}{x-4}\ge0

1.9 · Coordinate Plane & Circles

d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}

M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)

(x-h)^2+(y-k)^2=r^2

Mr. Azzam’s circle routine

Make the coefficients of $x^2$ and $y^2$ equal to 1.
Move the constant.
Group x-terms and y-terms.
Complete both squares.
Compare with standard form.

For diameter problems, midpoint first. For a circle tangent to an axis, radius equals the perpendicular distance from the center to that axis.

Quick Check

Check the idea before moving on.

What is the FIRST step for a circle whose diameter endpoints are given?

Mixed Method Selection · No lesson labels

Before calculating, identify the structure. This is the skill that separates chapter review from isolated lesson practice.

Ask yourself

What do I recognize? What is the first legal move? Which earlier rule controls this problem?

Do not guess from appearance

A radical may need a conjugate; a rational inequality needs a sign chart; repeated algebraic structure may suggest substitution.

Quick Check

Check the idea before moving on.

Choose the method FIRST.

\frac{x-1}{x+2}<3

Chapter 1 · Interactive Mastery

20 mixed questions: Warm-up → Core → Exam-style → Challenge. The question order deliberately mixes lessons so the method is not given away.

Curated problems written for this lesson. Select an answer, check it, and use the hint, full solution, or the AI Tutor if you get stuck — the tutor receives the question and concept as context.

Core practice first

Starting with Rational expressions at core practice level. Your current attempt stays in a fixed order; the next attempt adapts to new evidence.

0 of 20 answered

0 / 20Correct

Problem 1Core

Simplify using exponent rules.

x^{5/3}x^{-2/3}