FP2 §1 Inequalities

Question 1

What are critical values

Answer 1

Sign of an algebraic factor changes

Question 2

What extra care needs to be taken with inequality signs above standard ‘equals’ signs?

Answer 2

• The inequality sign can be treated as an equals sign as long as you definitely do not divide or multiply both sides of the expression by a negative number.
• In FP2 the expressions will be algebraic fractions

Question 3

Two solutions to the issue of possible negative numbers changing direction of inequality.

Answer 3

Solutions:

1. Don’t multiply through — subtract and find common denominators of algebraic fractions.
2. Multiply through only by even powers e.g. (x - 2)^2

Question 4

General strategy for solving inequality involving algebraic fraction.

Answer 4

1. Rearrange => Find the critical values
2. Use a sketch or table to identify solutions
3. Write down answers checking:
   (a) Whether inequalities are ‘strict’ or allow equality.
   (b) Whether any conflicts with original domain arise.

Question 5

What must be checked before writing down the solution to an inequality?

Answer 5

(a) Whether regions are ‘strict’ or allow equality.

(b) Whether any conflicts with original domain arise.

Question 6

What should you look for when sketching a graph of an expression in the inequality

Answer 6

• Try rearranging the inequality to make sketching easier
• Asymptotes can be made obvious
• Sketch the simplest modulus functions

Question 7

Adapt general strategy for modulus inequality

Answer 7

A sketch is usually the best approach if a modulus function is involved.
- Try to rearrange such that you only sketch the simplest/easiest function involving the modulus operator.

Question 8

Checking your solution for errors?

Answer 8

Check using calculator TABLE function and RTQ

Question 9

Best tactics for finding critical values?

Answer 9

• • Use method of multiplying by [f(x)]^2 (≥ 0, for all x ∈ ℝ)
• • … except where the common-denominator and simply approach is more convenient.