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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 or require conditions; squaring can introduce extraneous roots.

Learning objectives

  • Simplify, expand and factor expressions.
  • Solve equations/inequalities and check solutions.
  • Interpret slope and intercept of a line.

Prerequisite knowledge

  • Negative numbers, fractions and order of operations.

Core knowledge

Equivalence

Adding the same expression or multiplying by a nonzero number preserves an equation.

Gradient

In y = mx + c, m is Δy/Δx and c is the y-value at x = 0.

Domain

Rational expressions exclude zero denominators; real square roots require nonnegative radicands.

Worked example

3(x − 2) + 5 = 2x + 9 ⇒ 3x − 6 + 5 = 2x + 9 ⇒ x = 10. Substitution gives 29 on both sides.

Misconceptions to avoid

  • (a+b)² = a²+2ab+b², not a²+b².
  • Multiplying an inequality by a negative reverses its sign.

Virtual experiment procedure: Linear functions

  1. Define the unknown and its domain from context.
  2. Form the relation, transform line by line and justify risky steps.
  3. Substitute into the original equation and check the context.

Safety and cautions

  • Do not cancel a factor that may be zero without separating cases; always check domain restrictions.

Evidence to collect

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

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 5x − 7 = 18.

  1. x = 5
  2. x = 11/5
  3. x = −5
Show answer and explanation

Correct answer: x = 5

5x = 25, so x = 5.

2. Factor x² − 9.

  1. (x−3)(x+3)
  2. (x−9)(x+1)
  3. (x−3)²
Show answer and explanation

Correct answer: (x−3)(x+3)

Use the difference of two squares.

3. Solve −2x > 6.

  1. x < −3
  2. x > −3
  3. x < 3
Show answer and explanation

Correct answer: x < −3

Divide by −2 and reverse the inequality: x < −3.

Academic references

  1. 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.

Content publisher
WebLab STEM
Technical maintainer
Đức Tiến — Control & Automation Engineer