Unit 2 - Functions
Required Prior Knowledge
Questions
Sketch the following quadratics, labelling key points:
a) \(y=\left(x-2\right)\left(x+3\right) \)
b) \(y=-2\left(x+1\right)\left(x+6\right)\)
c) \(y=\left(x-2\right)^{2}-5\)
d) \(y=-\left(x+1\right)^{2}-4\)
e) \(y=x^{2}-2x-15\)
f) \(y=x^{2}+6x-16\)
Solutions
Get Ready
Questions
Solve the inequalities, and sketch the solutions on the number line.
a) \(3x+4\ge 16\)
b) \(2x-5\lt 3x+1\)
c) \(12-5x\gt 8\)
Solutions
Notes
To solve a quadratic inequality, we must first find the critical values.
These are the solutions (or roots) of the corresponding quadratic equation.
We then find where the graph is above (\(\gt\)) or below (\(\lt\)) the \(x\)-axis.
We can do this using a graph of the function or a sign diagram.
Examples and Your Turns
Example
Solve the inequality$$\left(x-3\right)\left(x+1\right)\ge 0$$
Your Turn
Solve the inequality$$\left(x+5\right)\left(x-2\right)\lt 0$$
Example
Solve the inequality$$3x^{2}+5x\ge 2$$
Your Turn
Solve the inequality$$x^{2}+9\lt 6x$$
Your Turn
Solve the inequality$$-2x^{2}-5x+3\le 0$$
Your Turn
Solve the inequality$$4-x^{2}\gt 0$$
Practice
Solve these inequalities, trying to do as little work as possible for each subsequent question.
1) \(x^{2}+4x+3\gt 0\)
2) \(x^{2}+4x+3\ge 0\)
3) \(x^{2}+4x+3\le 0\)
4) \(x^{2}+4x+3\lt 0\)
5) \(x^{2}+2x-3\lt 0\)
6) \(x^{2}+2x-3\gt 0\)
7) \(x^{2}-2x-3\gt 0\)
8) \(x^{2}-2x-8\gt 0\)
9) \(-x^{2}+2x+8\gt 0\)
10) \(-x^{2}+2x-8\le 0\)
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$$x\lt -3, x\gt -1$$
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$$x\le -3, x\ge -1$$
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$$-3\le x\le -1$$
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$$-3\lt x\lt -1$$
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$$-3\lt x \lt 1$$
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$$x\lt -3, x\gt 1$$
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$$x\lt -1, x\gt 3$$
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$$x\lt -2, x\gt 4$$
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$$-2\lt x\lt $$
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For all \(x\in\mathbb{R}\)
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
Explain the difference to the solution when the coefficient of \(x^{2}\) is negative.
Can a quadratic inequality have no solution? Explain.
Can a quadratic inequality have all real numbers as the solution? Explain.
Common Mistakes / Misconceptions
Over generalising and thinking that any inequality of the form \(ax^{2}+bx+c\lt 0\) will have solutions of the form \(p\lt x\lt q\). This is only true if \(a\gt 0\).
Incorrectly writing the final inequalities. For example, \(x\lt p\) or \(x\gt q\) are two separate regions and can’t be combined into \(p\gt x \gt q\).
Dividing (or multiplying) through by a negative and forgetting this changes the direction of an inequality. It is always safer to add and subtract to the other side.
Connecting This to Other Skills
You need to be able to solve quadratic equations (PK3) to solve quadratic inequalities, as well as be able to sketch quadratic functions (2.7).
When we look at the Discriminant (2.16) in the next skill we will need to be able to solve quadratic inequalities.
We look at more general inequalities, which build upon this idea in 2.24.
Self-Reflection
What was the most challenging part of this skill for you?
What are you still unsure about that you need to review?