Inequalities.md

A-Level Further Pure 1

Advanced Inequalities

When multiplying or dividing an inequality by a negative number you reverse the direction of the inequality sign so you need to be careful.

For a polynomial inequality move all terms to one side then factor/find roots to find critical values then use a sketch to determine correct intervals.

For an algebraic fraction you need to multiply by a positive number. You can do this by multiplying by a square number e.g.

$\frac{1}{x} < x \Rightarrow x < x^3 \neq 1 < x^2$

Using Graphs

If you can sketch graphs $y=f(x)$ and $y=g(x)$ you can solve the inequality $f(x)<g(x)$ by observing on the graph when each function is above/below the other. Still work out critical values by solving $f(x)=g(x)$ and use the graph to help with intervals.

Modulus Inequalities

When given $|f(x)| > g(x)$ you can simply solve by considering the 2 cases:

$f(x) > g(x)\wedge f(x) > 0$ $-f(x) > g(x) \wedge f(x) <0$

When given $|f(x)| < |g(x)|$ just consider the same 2 cases. or Find critical values where $f(x) = g(x)$ and $-f(x) = g(x)$ then sketch the graph

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Flashcards

#z_Legacy/Maths/A-Levels/Further-Pure/Inequalities, #Maths/Topics/Inequalites

How do you solve a polynomial inequality? ?

1. Move all terms to one side
2. Factor/find roots to find critical values
3. Use a sketch to determine correct intervals

What's the approach for solving an algebraic fraction inequality? ? Multiply by a positive number, such as a square number.

How can graphs be used to solve inequalities? ? Sketch $y=f(x)$ and $y=g(x)$, then observe where each function is above/below the other to solve $f(x)<g(x)$.

What are the two cases to consider when solving $|f(x)| > g(x)$? ? $f(x) > g(x)\wedge f(x) > 0$ $-f(x) > g(x) \wedge f(x) <0$