This course is about statements of the form less than or greater than, and the stretches of the number line that solve them. It starts with open, closed, half-open and infinite intervals, and how the words and and or turn into intersection and union. Eight rules then say exactly what may be done to an inequality and when the sign must be reversed. With them students solve linear inequalities, double inequalities and systems, then quadratic inequalities using the roots, and products and fractions of any degree by the wavy curve method. The course ends with the inequality between the arithmetic, geometric and harmonic means, used to find greatest and least values.
Middle Years
Intervals and Inequalities
3-week course.
- 7 lessons
Overview
Lessons
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Intervals on the real line
Open, closed and half-open intervals, infinite intervals, the length of an interval, and moving between interval notation, inequalities, set-builder form and the number line.
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Combining intervals with and and or
Union for or, intersection for and, the four cases built from the same two numbers, complements of intervals, and removing a point or a stretch.
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Rules for working with inequalities
Eight rules for inequalities: chaining, adding, changing signs, multiplying by positive or negative numbers, powers and reciprocals, with finding the range of an expression.
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Solving linear inequalities
Solving linear inequalities, reversing the sign when dividing by a negative number, double inequalities, systems of inequalities and everyday problems.
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Quadratic inequalities
Positive outside the roots and negative between them, quadratics with no real roots or equal roots, and using the discriminant for conditions on a coefficient.
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The wavy curve method
Solving inequalities with products and fractions by marking critical points and letting the sign wave, with repeated factors and points where a fraction is undefined.
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The A.M., G.M., H.M. inequality
The arithmetic, geometric and harmonic means of n positive numbers, the inequality A.M. ≥ G.M. ≥ H.M., and its use for greatest and least values.