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Mathematics · 2. Expressions, equations and inequalities

Algebra as a language of relationships

Translate situations into expressions, transform equivalently and check solutions in context.

40 minutesContent version: 2.1
Start the interactive pathwayOpen the virtual experiment

Driving question

When does an algebraic transformation preserve the solution set?

Curriculum coverage

Vietnam Mathematics 6–10; Cambridge IGCSE/AS Algebra.

Model scope and limitations

Dividing by a variable expression can lose solutions; cross-multiplying inequalities without sign verification flips relations invalidly; symmetric systems always require S² ≥ 4P.

Learning objectives

  • Solve higher-degree equations (biquadratic, symmetric product, reciprocal).
  • Apply 6 methods for irrational equations, 10 types of systems of equations, and 7 canonical inequality forms.
  • Optimize linear programming, parameter-dependent inequalities, and analyze linear gradients.

Prerequisite knowledge

  • Negative numbers, fractions and order of operations.

Core knowledge

Equivalence & Algebraic transforms

Adding the same expression or multiplying by a nonzero scalar preserves equations. Squaring may introduce extraneous roots; dividing by variable expressions can lose roots.

7 Canonical forms of inequalities & systems of inequalities

1. Linear & Quadratic (sign preservation on R: a·f(x)>0 ⇔ a>0, Δ<0); 2. Rational sign chart (odd-multiplicity flips sign, even keeps sign, denominator strictly open interval); 3. Absolute value (|f|≤g ⇔ −g≤f≤g; |f|≥g ⇔ f≥g or f≤−g); 4. Radical 4 equivalent rules (√f < g ⇔ f≥0, g>0, f<g²; √f > g ⇔ [g<0, f≥0] or [g≥0, f>g²]); 5. Exp & Log (reverse sign when 0<a<1); 6. Systems of inequalities (intersection S₁∩S₂ & convex polygon); 7. Parameter m optimization (m ≥ f(x) ∀x ⇔ m ≥ max f(x); m ≥ f(x) has solution ⇔ m ≥ min f(x)).

Special higher-degree & Biquadratic equations

Biquadratic ax⁴ + bx² + c = 0 uses t = x² (t ≥ 0). Symmetric products (x+a)(x+b)(x+c)(x+d) = m with a+b = c+d group into t = x² + (a+b)x. (x+a)⁴ + (x+b)⁴ = c uses t = x + (a+b)/2 to cancel odd powers.

6 Comprehensive methods for irrational equations

1. Power raising (√(A) = B ⇔ B ≥ 0, A = B²); 2. Complete substitution to systems; 3. Incomplete substitution (quadratic Δ_t); 4. Conjugate factoring (x − x₀); 5. Monotonic characteristic functions; 6. Inequality bounding (AM-GM, Cauchy-Schwarz, squares sum).

10 Comprehensive types of systems of equations

1. Linear Gauss & Cramer; 2. Linear-quadratic substitution; 3. Type 1 symmetric (S = x+y, P = xy, S² ≥ 4P); 4. Type 2 symmetric (subtract to factor x − y); 5. Homogeneous (divide by y² with t = x/y); 6. Factoring via quadratic Δ_x; 7. Monotonic functions & inequalities; 8. Irrational radical systems; 9. Exponential & Logarithmic systems; 10. Parameter m systems & integer solutions.

Worked example

Symmetric system: x + y = 5 and x² + y² = 13. Setting S = x+y, P = xy yields S² − 2P = 13 ⇒ P = 6. Since S² − 4P = 1 > 0, roots of t² − 5t + 6 = 0 are 2 and 3, giving pairs (2, 3) and (3, 2).

Misconceptions to avoid

  • (a+b)² = a²+2ab+b², not a²+b².
  • Multiplying an inequality by a negative reverses its sign.
  • Solving type 1 symmetric systems for S and P without checking S² ≥ 4P yields extraneous complex solutions instead of real roots.
  • Solving √(f(x)) = g(x) without checking g(x) ≥ 0 before squaring introduces extraneous roots.
  • In homogeneous systems, dividing by y² without checking y = 0 can discard valid solutions.
  • In biquadratics ax⁴ + bx² + c = 0, a double root t > 0 gives two real roots x = ±√t, not a double root in x.

Virtual experiment procedure: Linear functions

  1. Define the domain and constraints (non-negative radicand, nonzero denominator, t ≥ 0, S² ≥ 4P).
  2. Identify equation/system type (symmetric, homogeneous, conjugate, characteristic function) for optimal solution.
  3. Perform equivalent transforms, solve for unknowns and verify against constraints to eliminate extraneous roots.

Safety and cautions

  • Do not cancel a factor that may be zero without separating cases; always check domain and real-root existence conditions.

Evidence to collect

  • A solution with variable definition, equivalent steps and verification.
Open the virtual experiment
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Diagnostic check

1. Expand 2(x+3).

  1. 2x+6
  2. 2x+3
  3. x+6
Show answer and explanation

Correct answer: 2x+6

Multiply 2 by both terms.

2. The line y = −3x + 5 has what gradient?

  1. −3
  2. 5
  3. 3
Show answer and explanation

Correct answer: −3

The x coefficient in y = mx+c is m.

Application practice

1. Solve biquadratic equation x⁴ − 5x² + 4 = 0.

  1. x ∈ {−2, −1, 1, 2}
  2. x ∈ {1, 4}
  3. x ∈ {−4, −1, 1, 4}
Show answer and explanation

Correct answer: x ∈ {−2, −1, 1, 2}

Setting t = x² ≥ 0 gives t² − 5t + 4 = 0 ⇔ t = 1 or t = 4. Hence x = ±1 and x = ±2, yielding 4 real solutions.

2. How many real pairs (x, y) solve x + y = 6 and xy = 8?

  1. 2 pairs: (2, 4) and (4, 2)
  2. 1 pair: (3, 3)
  3. 0 pairs
Show answer and explanation

Correct answer: 2 pairs: (2, 4) and (4, 2)

With S = 6 and P = 8, S² − 4P = 4 > 0. The quadratic t² − 6t + 8 = 0 has roots 2 and 4, producing the two pairs (2, 4) and (4, 2).

3. What is the real solution set of √(x − 1) = 3 − x?

  1. {2}
  2. {2, 5}
  3. {5}
Show answer and explanation

Correct answer: {2}

Condition 3 − x ≥ 0 ⇔ x ≤ 3. Squaring gives x − 1 = (3 − x)² ⇔ x² − 7x + 10 = 0, giving x = 2 and x = 5. Because x ≤ 3, x = 5 is extraneous, leaving only x = 2.

Academic references

  1. Cambridge IGCSE Mathematics 0580 — official syllabus and resources
  2. Cambridge AS & A Level Mathematics 9709 — official syllabus
  3. OpenStax Precalculus 2e — functions, geometry and probability

This lesson is maintained against the cited sources below. Simulations are learning models and do not replace supervised physical-laboratory safety procedures.

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WebLab STEM
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Đức Tiến — Control & Automation Engineer