Chapter 0: Inequalities

Question 1

Week 1

State the rules to follow while solving inequalities involving the mod (|x|) operation.

Answer 1

The rules are as follows:

• If|x|< a, then -a < x < a
• If |x|> a then x > a OR x < -a
• If |x| < -a, the solution set does not exist
• If |x| > -a, the solution set contains all real numbers.

Question 2

Week 1

State how to solve inequalities involving two mod operations i.e. |a| > |b|

Answer 2

You can follow one of two methods to solve these types of inequalities.

The Graphing Method

You simply graph both absolute value functions and deduce the interval over which the value of |a| is greather than |b|.

The Calculation Method

You convert the absolute value functions into polynomials by squaring both sides of the inequality

|a| > |b| —-> (a)² > (b)²

You can then expand both sides and solve the inequality as you would a regular quadratic inequality.

Question 3

Week 1

State how to solve a quadratic inequality.

Answer 3

Let’s take an example

4x² - 12x + 9 > x² + 6x + 9

Step 1: Combine Like Terms

3x² - 18x > 0

Step 2: Simplify & Find Roots of The Polynomial

x(x-6) > 0

Represent the Critical Points Using a Table

Do this in order to find out the interval over which the polynomial is greater than 0. The interval you find is the solution set of your inequality.