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Mathematics I · Week 3 — Equations and algebraic reasoning

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Mathematics I · Week 3 — Equations and algebraic reasoning

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2026-07-19

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Week 3 — equations and algebraic reasoning

Algebra is a chain of permissions. Each line should preserve the stated claim or clearly record a new condition. The goal is not to make symbols look simpler; it is to preserve the solution set.

The audit question

At every arrow, ask: “Is this step reversible for every currently allowed value?” If not, record the condition and check candidates in the original statement.

Learning objectives

By the end of this week, you should be able to:

Separate solution-preserving transformations from moves that only create candidates.
Keep restrictions visible when solving rational, radical, and squared equations.
Use factorisation, branches, and substitution to verify a final solution set.
Solve inequalities with a sign chart and include only valid endpoints.
Translate absolute value into distance before choosing cases.

Prerequisites

Week 2 domain restrictions, particularly forbidden denominators and real square roots.
Factoring simple polynomials and reading interval notation on a number line.
The idea that x² = a can have two real solutions when a > 0.

Core content

Equations and equivalence

An equation asks which allowed inputs make two expressions equal. An equivalent transformation preserves exactly the same solution set. Adding the same quantity to both sides is reversible. Dividing by an expression is reversible only when that expression is known non-zero. Squaring can create additional candidates because different values can have the same square.

This is why the final line is not the end: substitute every candidate into the original equation. The original statement, not a transformed version, is the test of validity.

Step	Usually safe?	Condition to remember
Add/subtract the same expression	yes	expression is defined
Multiply/divide by an expression	only conditionally	divisor must be non-zero
Square both sides	may add candidates	test in original equation
Take a square root	requires branches	`x²=a` has `±√a` when a is positive
Cancel a factor	only conditionally	cancelled factor may be zero

Factorisation and cases

The zero-product property says that if ab = 0, then a = 0 or b = 0. It does not say both are zero. Factorisation often exposes cases; write each case separately rather than cancelling the factor that contains information.

For an inequality, sign matters. Multiplying or dividing by a negative reverses the inequality sign. If the sign of an expression is unknown, split into cases or use a sign chart rather than performing a blanket division.

Original worked example — candidates are not conclusions

Suppose an equation contains a square root. Squaring may be a useful move, but it only gives candidates, because both positive and negative quantities square to the same result. After solving the squared equation, substitute every candidate into the original expression. A candidate that violates the original side condition is extraneous, not an answer.

Worked audit — one valid candidate, one extraneous candidate

Solve √(x+1)=x-1 over the reals. The left side is non-negative, so the right side must be non-negative too: record x≥1 before squaring. Squaring gives

x+1=(x-1)²=x²-2x+1, so x²-3x=0, and the candidates are x=0 and x=3.

The restriction already rejects 0. Direct substitution confirms the remaining candidate: √(3+1)=2 and 3-1=2. Therefore the solution set is {3}, not {0,3}. The step “square both sides” gave an implication from the original equation to the squared equation; it did not make the two statements automatically equivalent.

Write a candidate marker

After any non-reversible step, use ⇒ in your scratch work rather than ⇔. It is a small visual reminder to test the final values in the original condition.

Exponents, roots, and absolute value

Exponent rules depend on their conditions. For example, √(a²) = |a|, not automatically a. Square roots represent non-negative principal roots, while an equation such as x² = 9 has two real solutions. An absolute-value equation describes distance; |x-a| = r has two branches when r > 0, none when r < 0, and one when r = 0.

A reliable solution protocol

State restrictions from denominators, roots, or context.
Choose a transformation and name why it is valid.
Split branches when a product, absolute value, or squared relation requires it.
Retain conditions created by a conditional step.
Check every candidate in the original equation or inequality.

Inequalities as sign maps

An inequality asks where an expression is positive, negative, or zero. Factorisation gives critical points; those points split the number line into intervals. Test one value per interval and then decide whether endpoints are included by the original relation (>, ≥, <, or ≤). This is more reliable than treating an inequality as an equation and guessing a direction.

For absolute value, think distance. |x-a| = r asks for points distance r from a; it normally creates two branches. An inequality such as |x-a| < r describes an interval of points close enough to a.

Worked sign chart — intervals carry the answer

Consider ((x-1)(x+2))/(x-3) ≥ 0. The critical values are -2, 1, and 3: the numerator is zero at -2 and 1; the denominator is undefined at 3. They divide the real line into four intervals.

Interval	Sign of `(x-1)(x+2)`	Sign of `x-3`	Overall sign
`(-∞,-2)`	positive	negative	negative
`(-2,1)`	negative	negative	positive
`(1,3)`	positive	negative	negative
`(3,∞)`	positive	positive	positive

The relation is ≥0, so include the numerator zeros -2 and 1; exclude 3 because the original expression is undefined there. The solution is [-2,1] ∪ (3,∞). Notice that the answer is made of intervals, not merely the critical values. A sign chart is a map of what happens between the points where a sign can change.

Absolute value: translate distance before branching

|x-4|=3 means “the distance from x to 4 is 3,” giving x=1 or x=7. In contrast, |x-4|≤3 means “the distance from x to 4 is at most 3,” giving the full interval [1,7].

The right side controls the case split:

|u|=r, with r<0: no solution, because distance cannot be negative.
|u|=0: exactly the values where u=0.
|u|=r, with r>0: solve u=r or u=-r.

Do not turn an absolute-value inequality into two equalities. The words “within,” “at most,” and “less than” describe an interval; “exactly distance r” describes boundary points.

See roots become geometry

A quadratic does not merely produce symbols in a formula. Its discriminant predicts how many times the graph meets the x-axis: positive gives two real intersections, zero gives one repeated touch, and negative gives none. In the lab, vary one coefficient at a time. Before releasing the slider, predict whether the number of real roots will change and identify what the graph must do for that to happen.

Interactive Simulation
Quadratic Function — f(x) = ax² + bx + c
f(x) = 1x² + 0x + -4
aopens up if +1
bhorizontal shift0
cy-intercept-4
📍 Vertex: (0, -4)
📐 Axis of Symmetry: x = 0
△ Discriminant (b²-4ac): +16 (2 real roots)
🎯 Roots: 2, -2
Loading Visualizer...

Connect the representation

The same value, b² − 4ac, is an algebraic discriminant and a graph-intersection count. That does not replace checking the conditions a ≠ 0 and real-number context; it gives you a second way to audit the answer.

Key formulas and translation sheet

Idea	Compact form	Read it as
Equivalence	`statement₁ ⇔ statement₂`	Each statement has exactly the same solutions.
Candidate-producing step	`statement₁ ⇒ statement₂`	Every old solution survives, but new candidates may appear.
Zero product	`ab = 0 ⇒ a = 0 or b = 0`	Split the factor cases; do not cancel the useful factor.
Principal square root	`√(a²) =	a
Absolute-value equation	`	u
Quadratic discriminant	`b² - 4ac`	Positive, zero, or negative predicts two, one, or no real roots.

Common pitfalls

Cancelling can discard an entire case

When a factor could be zero, cancelling it loses that branch. Use the zero-product property or write explicit cases before reducing the expression.

A square can hide a sign

From x² = 9, both x = 3 and x = -3 work. Conversely, √(x²) must be |x|, because the principal square root is non-negative.

Inequalities remember the sign of your multiplier

Multiplying or dividing by a known negative reverses < and >. If the sign is unknown, make a sign chart or split cases instead of applying one rule blindly.

A candidate is not yet a solution

Squaring, clearing denominators, and other conditional steps can enlarge the candidate set. Substitute each remaining candidate into the original statement.

Critical points are boundaries, not the complete answer

For an inequality, test the intervals between critical values. Then return to the original relation to decide whether a zero is included or a denominator zero is excluded.

Retrieval ladder

Name a step that preserves solutions, one that needs a non-zero condition, and one that can create extra candidates.
Explain why √(x²) is |x|, not always x.
Describe why a sign chart needs intervals rather than only critical points.
State the one check that turns a candidate into a verified solution.
In the quadratic lab, create one graph with two roots, one with a repeated root, and one with no real roots. For each, state the sign of the discriminant before reading it.

Five-minute exit check

You are ready when you can annotate an algebraic line with its permission and can explain why the original equation—not the prettiest transformed form—has the final say.

Retrieval practice

Write down one algebraic operation that is always reversible, one that needs a non-zero condition, and one that may introduce candidates. For each, explain what check prevents a false final answer.