Solving Polynomial Inequalities

Introduction

So the approach for this needed a picture. Maybe because of my brain.

He uses an open circle on the number line to denote less than. He would use a closed circle if less than or equal to. He basically solved the question as shown which is not easy but it was the bit after

• He draw a number line
• Put the values obtained on it
• Then said there were 3 area, less -5 -> -∞, between -5 and 2 and greater than 2 -> +∞
• He plugged in numbers from each section demonstrating
  • if you choose the left you always get a +
  • if you choose the middle you always get a -
  • if you choose the right you always get a +

There were two ways to write the answer using

Interval Notation<

(-∞, -5) ∪ (2,+∞)

or as an inequality

x > 2 or x < -5

Example 2

So the next example was

  x³ - x <= 3x² -3 
  x³ - 3x² - x + 3 <= 0
  x²(x-3) -1(x-3) <= 0
 (x-3)(x² -1) <= 0
 (x-3)(x+1)(x-1) <= 0

So three factors.

This example demonstrated that the positive and negative areas always alternate. He also showed that if using interval notation round brackets mean less than and square brackets mean less than or equal to.