Unit 2 - Functions
Required Prior Knowledge
Questions
Solve the following inequalities
a) \(3x-5\gt7\)
b) \(x^{2}-x-6\le 0\)
Solutions
Get Ready
Questions
Solve $$x^{3}-x^{2}-10x-8=0$$Hence sketch the graph of $$y=x^{3}-x^{2}-10x-8$$
Solutions
Notes
Shown is the graph of the function \(y=f\left(x\right)\).
We can see from the graph that \(f\left(x\right)\gt 0\) when \(-1\lt x\lt 2\) or \(x\gt 3\) (as these are the \(x\) values for which the graph is above the \(x\)-axis and therefore positive).
We also see that \(f\left(x\right)\lt 0\) when \(x\lt -1\) or \(2\lt x \lt 3\) (as these are the \(x\) values for which the graph is below the \(x\)-axis and therefore negative).
If given the equation and not the graph, we need to find the critical points.
We can do this by solving the equation or by using our GDC.
We can then either sketch the graph or construct a sign diagram.
Examples and Your Turns
Example
Solve $$x^{3}-x^{2}-10x-8\ge 0$$
Your Turn
Solve $$x^{3}+4x^{2}+x-6\gt 0$$
Your Turn
Solve $$2x^{3}-5x^{2}-6x+4\lt 0$$
Your Turn
Solve $$1-4x^{2}\lt 5x^{3}+4x$$
Notes
For inequalities of the form\(\frac{f\left(x\right)}{g\left(x\right)}\gt 0\) we have to consider both the numerator and the denominator by considering cases.
When \(f\left(x\right)\gt 0\) and \(g\left(x\right)\gt 0\): the whole fraction is positive.
When \(f\left(x\right)\gt 0\) and \(g\left(x\right)\lt 0\): the whole fraction is negative.
When \(f\left(x\right)\lt 0\) and \(g\left(x\right)\gt 0\): the whole fraction is negative.
When \(f\left(x\right)\lt 0\) and \(g\left(x\right)\lt 0\): the whole fraction is positive.
Examples and Your Turns
Example
Solve$$\frac{x-1}{x+2}\gt 3$$
Your Turn
Solve$$\frac{x+3}{x-1}\le 2$$
Your Turn
Solve$$\frac{x+1}{x-2}\le x$$
Notes
There are three approaches to solving inequalities involving the modulus function.
Like all inequalities we could graph it and find regions that way.
The second way is to split the inequality into two pieces using the rules$$\left|a\right|\lt k\iff -k\lt a\lt k\\\left|a\right|\gt k\iff a\lt -k\text{ OR }a\gt k$$
If both sides of the inequality are positive then we can square both sides.
Examples and Your Turns
Example
Solve $$\left|2x-1\right|\lt 3$$
Your Turn
Solve $$\left|4-x\right|\ge 5$$
Your Turn
Solve $$\left|2x-4\right|\lt 4$$
Your Turn
Solve $$\left|3-2x\right|\ge 1$$
Example
Solve $$\left|3x+1\right|\ge \left|2x-3\right|$$
Your Turn
Solve $$\left|\frac{x}{x+1}\right|\ge \frac{1}{2}$$
Your Turn
Solve $$\left|x^{2}-4x\right|\le 3x-5$$
Your Turn
Solve $$\left|x^{2}-5x+3\right|\gt 3$$
Key Facts
Use this applet to generate a prompt for a Key Fact that you need to know for the course. The idea is that you should KNOW these key facts in order to be able to solve problems.
Taking it Deeper
Conceptual Questions to Consider
Which method do you prefer for solving inequalities once you know the critical values: sketch a graph or table of values? Explain the difference and why you have a preference.
Explain why a root of the denominator of a rational inequality can never be a solution to the inequality.
Explain the reasoning behind the four cases for solving rational inequalities.
Sketch on a number line the solution sets to \(\left|x\right|<3\) and \(\left|x\right|>3\). How do these link to the idea of ‘distance to a point’?
Common Mistakes / Misconceptions
When solving rational inequalities, we can never multiply by the denominator (like we would for an equation) as for some values of \(x\) the denominator will be negative which would ‘flip’ the inequality.
Forgetting to remove extra solutions if you square an inequality involving absolute value functions, or doing this when one side is (or could be) negative.
Connecting This to Other Skills
This skill builds on the idea of solving equations (2.23) and takes it a little bit further.
Understanding Quadratic Inequalities (2.15) is essential to understand inequalities with other functions.
Self-Reflection
What was the most challenging part of this skill for you?
What are you still unsure about that you need to review?