Equations and inequalities
What is an equation
What an equation is and how it is defined. The left-hand and right-hand sides, the graphical meaning and the number of solutions: none, one, several or infinitely many.
Solving equations
Which changes to an equation keep it true. Adding, multiplying, dividing and raising to a power, and the traps that more complicated expressions bring.
Linear equations
How to solve a linear equation of the form ax+b=0. The method of equivalent steps, the geometric meaning and worked examples, including a word problem.
Quadratic equations
How to solve quadratic equations: those with no x term, those with no constant term and the general case using the discriminant. With step-by-step methods.
Biquadratic equations
How to solve a biquadratic equation by using a substitution that turns it into a quadratic equation. The method explained step by step with examples.
Binomial equations
What a binomial equation is, its general form and how to solve it. Making the leading coefficient 1 and finding the roots in the complex numbers.
Linear inequalities
How to solve linear inequalities using equivalent steps. Ordinary inequalities, inequalities with the unknown in the denominator, and inequalities with a modulus.
Quadratic inequalities
How to solve quadratic inequalities both algebraically and graphically. The method depends on the sign of the coefficient a and on the value of the discriminant of the equation.
Inequalities in product and quotient form
How to solve inequalities in product and quotient form using critical values. Signs in the intervals, even powers and why you should not multiply by the denominator.
Exponential and logarithmic inequalities
How to solve exponential and logarithmic inequalities. When the inequality sign flips, restrictions on logarithms and worked examples.
Exponential equations
How to solve exponential equations: by writing both sides with the same base, by taking logarithms or by substitution. All three methods on worked examples.
Logarithmic equations
How to solve logarithmic equations: from the definition, by removing the logarithms, with the laws of logarithms and by substitution. With restrictions, checks and worked examples.
Equations with square roots
How to solve equations with the unknown under a square root. Squaring, the compulsory check, getting the root on its own, and equations with two roots.
Trigonometric equations
How to solve trigonometric equations directly and by substitution. The basic form of the equation, the formulae you need and worked examples.
Trigonometric inequalities
How to solve trigonometric inequalities with sine, cosine and tangent. Solving on the unit circle, writing intervals with a period and the most common mistakes.
Simultaneous equations
How to solve simultaneous linear equations. The substitution and elimination methods step by step, the number of solutions and worked examples.
