Chapter 3.3 Solving Piecewise Rational Inequalities

In this section we will discuss how to solve inequalities involving polynomials.

Polynomial Inequalities

To solve a polynomial inequality of the form \[f(x) > 0,\] write the polynomial in factored form and sketch it.

The \(x\)-values of points on \(f\) that are above the \(x\)-axis are the solutions to the inequality.

Of course, factoring polynomials is a difficult problem. If a polynomial contains a positive irreducible quadratic factor, then this factor does not change the roots or the solutions of the inequality.

In determining solutions to inequalities, such factors may be discarded.

Note: An irreducible quadratic polynomial is a quadratic polynomial that has no real roots. It turns out that any polynomial is a product of linear and irreducible quadratic factors.

Example 1

Find all solutions to the inequalities

  1. \((x+2)^3(x-2)^2(x-3) > 0\),
  2. \((x+2)^3(x-2)^2(x-3) \ge 0\),
  3. \((x+2)^3(x-2)^2(x-3) < 0\),
  4. \((x+2)^3(x-2)^2(x-3) \leq 0\).

Graph \(f(x)=(x+2)^3(x-2)^2(x-3)\).

  1. \((-\infty,-2)\cup(3,\infty)\)
  2. \((-\infty,-2]\cup\{2\}\cup[3,\infty)\)
  3. \((-2,2)\cup(2,3)\)
  4. \([-2,3]\)

In this example, we split up the compound inequality to solve this polynomial inequality.

Example 2

Find all solutions to the compound inequality \[2\le x^2-2x+2 < 10.\]

Solve \(2\le x^2-2x+2\) and \(x^2-2x+2<10\).

The inequality \(2\le x^2-2x+2\) is the same as \(0\le x^2-2x\). Factor and graph it to get the solution is \((-\infty,0]\cup[2,\infty)\)]

The inequality \(x^2-2x+2<10\) is the same as \(x^2-2x-8<0\). Factor and graph it to get the solution is \((-2,4)\)

The solution is the intersection of the two solution sets: \((-2,0]\cup[2,4)\).

Inequalities Involving Rational Functions

As with polynomials, sketching a rational function, \(f\), immediately determines the solution sets to the rational inequalities \[f(x) > 0, \quad f(x) \ge 0, \quad f(x) < 0, \quad {\rm and}\quad f(x) \leq 0.\]

We need only observe where \(f\) lies above and below the \(x\)-axis, and where it intersects the \(x\)-axis.

Example 3

Suppose that \[f(x) = \dfrac{(x+3)^2(x+1)(x-3)^5}{x(x-5)^2(x+2)^3}.\] Find all solutions to

  1. \(f(x) > 0\),
  2. \(f(x) \ge 0\),
  3. \(f(x) < 0\),
  4. \(f(x) \leq 0\).

The graph of \(f(x)\) looks like this

  1. \((-\infty,-3)\cup(-3,-2)\cup(-1,0)\cup(3,5)\cup(5,\infty)\),
  2. \((-\infty,-2)\cup[-1,0)\cup[3,5)\cup(5,\infty),\)
  3. \((-2,-1)\cup(0,3)\),
  4. \(\{-3\}\cup(-2,-1]\cup(0,3]\).

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