A polynomial inequality occurs when a polynomial is less than zero or less than or equal to zero, greater than zero or greater than or equal to zero. Solving a polynomial inequality is similar to solving a quadratic inequality. The inequality does not equal for expression. We are solving the polynomial inequalities by replacing the in-equal symbol in expression. The polynomial inequality is solved by the factorization and substitution. The interval values are used in polynomial inequalities. Example for polynomial inequalities is x^{2} + 17 x + 19 > 0, x + 2 > 0.

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**Below are steps for solving polynomial inequalities:**

Step 1: Find the zeros of the polynomial. Make sure that zero is one side of the inequality, and then factor the polynomial.

Step 2: Create a sign chart to solve the polynomial inequality.

Step 3: Evaluate the sign of the polynomial at each of test values and to determine which region satisfy the inequality.

Step 4: The solution to a polynomial inequality will include the zeros of the function if the inequality is $\geq$ or $\leq$. If the inequality is strict (means > or <) we use parentheses next to the zeros of the polynomial.

Examples for Solving Polynomial Inequalities:

The polynomial is an equation with degrees and the inequalities values are substitute for variable determination. The inequality may be represented with symbols <, > or =.

**Example 1:**

Solve the quadratic inequality equation $x^2 - 7x - 18 > 0$

**Solution:**

The given quadratic inequality equation is $x^2 - 7x - 18 > 0$

The given equation is in the inequality equation. Change the in- equal symbol as the equal symbol. Determine the solution for the equality equation.

$x^2 - 7x - 18 > 0$ is converted into $x^2 - 7x - 18 = 0$.

Find the factors for that equation.

$x^2 - 7x - 18 = x^2 - 9x + 2x - 18$

= $x^2 - 9x + 2x - (9 \times 2)$

Take x as common in first two terms and take 2 as common in the next two terms.

$x^2 - 7x - 18 = x(x - 9) + 2(x - 9)$

= (x - 9) (x + 2)

Equate x - 9 and x + 2 to 0 to get the roots for the given equation.

x - 9 = 0 and x + 2 = 0

x = 9 and x = - 2

The roots for the equation $x^2 - 7x - 18 = 0$ are x = 9 and x = - 2.

Perform the sign analysis:

Put x = 0 => $x^2 - 7x - 18 = - 18 > 0$ ( False)

Put x = - 3 => $x^2 - 7x - 18 = 12 > 0$ (True)

Put x = 10 => $x^2 - 7x - 18 = 12 > 0$ (True)

So the solution for the inequality equation $x^2 - 7x - 18 > 0$ is x > 9 and x < - 2.

**Example 2:**

Solve the linear inequality equation x - 54 > 0

**Solution:**

The given linear inequality equation is x - 54 >0

The given equation is in the inequality equation. The equal symbol is replaced for in-equal symbol. The variable is determined as,

x - 54 = 0

Add 42 on both sides of the equation.

x - 54 + 54 = 0 + 54

x = 54

Convert equality into the inequality.

So the value is changed to x > 54.

The value for x - 54 > 0 is x > 54.

Practice Problem for Solving Polynomial Inequalities:

1. Solve the inequality equation x + 91 > 0.

Answer: x > - 91.

2. Solve the inequality equation $x^2 + 10 x + 21 > 0$

Answer: x > 7 and x > 3.