Welcome to Chapter 4, Part 4!

Hello Math Students! In today's session, we will be tackling the exciting challenge of solving inequalities. We'll explore various types, including quadratic, polynomial, and rational inequalities. Grab your pencils, and let's get started!

Topic List:

  • Solving a quadratic inequality written in factored form
  • Solving a quadratic inequality
  • Solving a polynomial inequality: Problem type 1
  • Solving a polynomial inequality: Problem type 2
  • Solving a polynomial inequality: Problem type 4
  • Solving a rational inequality: Problem type 1
  • Solving a rational inequality: Problem type 2

Solving Quadratic Inequalities in Factored Form

Let's start with quadratic inequalities already conveniently factored. The key here is to find the critical points and test intervals.

Example: Solve $(x-1)(x-6) < 0$

  1. Find Critical Points: Set each factor to zero: $x-1=0$ and $x-6=0$, giving $x=1$ and $x=6$.
  2. Create a Number Line: Mark these critical points on a number line. These points divide the number line into intervals.
  3. Test Intervals: Choose a test value in each interval and plug it into the original inequality.
    • For $x < 1$, test $x=0$: $(0-1)(0-6) = (-1)(-6) = 6 > 0$ (False)
    • For $1 < x < 6$, test $x=2$: $(2-1)(2-6) = (1)(-4) = -4 < 0$ (True)
    • For $x > 6$, test $x=7$: $(7-1)(7-6) = (6)(1) = 6 > 0$ (False)
  4. Write the Solution: The interval where the inequality is true is $(1, 6)$.

Solving General Quadratic Inequalities

Sometimes, you'll need to factor the quadratic yourself.

Example: Solve $x^2 - 2x \ge 8$

  1. Rearrange to Standard Form: $x^2 - 2x - 8 \ge 0$
  2. Factor: $(x-4)(x+2) \ge 0$
  3. Find Critical Points: $x=4$ and $x=-2$
  4. Test Intervals:
    • For $x < -2$, test $x=-5$: $(-5-4)(-5+2) = (-9)(-3) = 27 > 0$ (True)
    • For $-2 < x < 4$, test $x=0$: $(0-4)(0+2) = (-4)(2) = -8 < 0$ (False)
    • For $x > 4$, test $x=5$: $(5-4)(5+2) = (1)(7) = 7 > 0$ (True)
  5. Write the Solution: $(-\infty, -2] \cup [4, \infty)$

Solving Polynomial Inequalities

The same principles apply to higher-degree polynomials. Find the roots and test intervals.

Example: Solve $(3-x)(x+1)(x+5) < 0$

  1. Find Critical Points: $x=3$, $x=-1$, $x=-5$
  2. Test Intervals: (Remember to consider the sign changes!)
  3. Write the Solution: $(-5, -1) \cup (3, \infty)$

Solving Rational Inequalities

Rational inequalities involve fractions. Make sure to also consider where the denominator is zero, as these points are excluded from the solution.

Example: Solve $\frac{-x-1}{x-7} \ge 0$

  1. Find Critical Points: Set the numerator and denominator to zero: $-x-1 = 0$ and $x-7 = 0$, giving $x=-1$ and $x=7$.
  2. Test Intervals:
  3. Write the Solution: Remember that $x=7$ is not included because the denominator cannot be zero. The solution is $[-1, 7)$.

Keep practicing, and you'll master these inequalities in no time! Good luck with your studies!