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  5. Solving Inequalities (Graphing)

Solving Inequalities (Graphing)

Solving Polynomial Inequalities by graphing involves using algebraic techniques to determine the intervals where the polynomial is positive or negative and then graphing the solution.

Steps for Solving Inequalities

  1. Move all terms to one side and make sure the polynomial is in Standard Form
  2. Identify Critical Points of the polynomial by setting it equal to \(0\). This can be found by factoring the polynomial
  3. Graph the polynomial on the points where its \(0\), then graph additional points in between to determine if the polynomial is positive or negative at that point
  4. Write the solution to the inequality based on the intervals where the polynomial is positive or negative

Example

Consider the polynomial inequality \(P(x) = x^3 - 3x^2 - 4x + 12 > 0 \).

Since the polynomial is already written in descending order of degree, we won't need to set things up any further.

In order to identify Critical Points, we can first set the polynomial equal to 0:

\(P(x) = x^3 - 3x^2 - 4x + 12 > 0 \)

Next, we can fully factor the polynomial to determine its roots:

\(P(x) = x^2(x - 3) -4(x -3) = 0\)

\(P(x) = (x^2 -4)(x-3) = 0\)

\(P(x) = (x+2)(x-2)(x-3) = 0\)

Therefore, we can determine the Critical Points as \(x = -2, 2, 3\). These represent the points where \(P(x) = 0\).

We can sketch a graph of the function to illustrate where its greater than \(0\):

Graph of cubic function, x³-3x²-4x+12, showing where its greater than 0.

Therefore, we can determine that the solution to the inequality \(P(x) > 0\) is \(\boldsymbol{x \in (-\infty, -2) \cup (2, 3)}\).


Solve the following inequalities.

\(f(x)= x^2 - 10 \lt 3x \)

First, we can move all the terms onto one side and arrange them in descending order of degree:

\(f(x) = x^2 -3x - 10 \lt 0 \)

In order to identify the polynomial's Critical Points, we can set it equal to \(0\):

\(f(x) = x^2 -3x - 10\)

Next, we can fully factor the polynomial:

\(f(x) = (x-5)(x+2)\)

We can determine the Critical Points as \(x = -2, 5\).

We can sketch a graph of the function to illustrate where it's less than \(0\):

Graph of quadratic function, x²-3x-10, showing where its less than 0

Therefore, we can determine that the solution to the inequality \(f(x) = x^2 - 10 \lt 3x\) (or \(f(x) = x^2 - 3x - 10 \lt 0\)) is \(\boldsymbol{x \in (-2, 5)}\).

\((x+1)(x-3)^2 \gt 0 \)

Since the polynomial is already fully factored, we won't need to set things up any further.

We can determine the roots as \(x = -1, 3\).

We can sketch a graph of the function to illustrate where its greater than \(0\):

Graph of cubic function, (x+1)(x-3)², showing where its greater than 0.

Therefore, we can determine the solution to the inequality \(P(x) \gt 0\) is \(\boldsymbol{x \in (-1, 3) \cup (3, \infty)}\).

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