Solving Polynomial and Rational Inequalities

.

Sol 1 Factoring gives or . Marking 0,3, and -2 on a number line, and using that

and that all the exponents are odd, we get the sign chart shown below:

Therefore the solution is given by .

Sol 2 Factoring gives or . Marking 0,3, and -2 on a number line, and using that

and that the sign changes at 3 and -2 but does not change at 0, we get the sign chart shown below:

Therefore the solution is given by .

Sol 3 Factoring gives

and marking 4,1,2, and -2 on a number line, and using that

and that all the exponents are odd, we get the sign chart shown below:

Since the inequality is not strict, we can include the zeros of the numerator; so the solution is given by .

Sol 4 Factoring gives

and marking 5/2,-2, 4, and -3 on a number line, and using that

and that all the exponents are odd, we get the sign chart shown below:

Therefore the solution is given by .

Sol 5 Factoring gives

or

Marking -3,-2,-1,1,2, and 3 on a number line, and using the facts that

and that all the exponents are odd, we get the following sign chart:

Since the inequality is not strict, we can include the zeros of the numerator; so the solution is given by .

Sol 6 Since for all , and therefore for all ; so multiplying by gives the equivalent inequality .

Factoring yields or ; so marking on a number line and using that

when and all the exponents are odd, we get the following sign chart:

Therefore the solution is given by .

Sol 7 Factoring gives the inequality

or

Marking off -2,0,1/3,1,3/2, and 4, and using the facts that

and the sign changes at 3/2,1,-2, and 1/3 and does not change at 0 or at 4, we get the following sign chart:

Since the inequality is not strict, we can include the zeros of the numerator; so the solution is given by .
Correction:  the solution should include x=0 as well.

Sol 8 Subtracting from both sides gives , so or .

Marking -4,0,and 4 on a number line, and using that

and that all the exponents are odd, we get the following sign chart:

Therefore is the solution.

Sol 9 Subtracting from both sides gives

so

Therefore

so

or

Marking 2,4, and -1 on a number line, and using the facts that

and that all the exponents are odd, we get the following sign chart:

Therefore is the solution.