Chapter 1: Equations and Inequalities

An equation for me has no meaning unless it expresses a thought of God.
— Srinivasa Ramanujan

Why This Matters

An equation is a statement of balance. Two expressions, equal in value.

The art of solving equations is the art of maintaining that balance while isolating what you seek. It’s detective work — following clues, making logical deductions, arriving at truth.

Sections

Section Topic Key Skill Status

1.1

Linear Equations

Solve \(ax + b = c\)

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1.2

Applications of Linear Equations

Word problems

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1.3

Complex Numbers

Work with \(i = \sqrt{-1}\)

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1.4

Quadratic Equations

Factor, complete square, formula

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1.5

Applications of Quadratics

Motion, area, optimization

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1.6

More Equations

Polynomial, radical, rational

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1.7

Linear and Compound Inequalities

Solve and graph

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1.8

Absolute Value Equations/Inequalities

stem:[

x

= a], stem:[

x

< a]

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Mathematical Wisdom

The quadratic formula:

\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

This single formula solves every quadratic equation. It was known to Babylonian mathematicians 4,000 years ago (in geometric form), yet it remains as powerful today as it was then.

The discriminant \(b^2 - 4ac\) tells a story:

  • Positive: Two real solutions (the parabola crosses the x-axis twice)

  • Zero: One real solution (the parabola touches the x-axis)

  • Negative: Two complex solutions (the parabola never touches the x-axis)

One formula. Infinite equations. Eternal truth.

Study Tip

When solving equations, check your solutions by substituting back into the original equation. This catches extraneous solutions introduced by squaring or multiplying by variables.

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