Review on solving inequalities from College Algebra

Qudadratic Inequalities

Type I. Standard Form (coefficients of x are positive and the degree of each factor is 1.)

Example: Solve (x-2)(3x+1)<0. We get two points tex2html_wrap_inline29 and 2 on the real line as if we have equality, and we alternate + and - from the right to the left, so our answer to this question is tex2html_wrap_inline37

Type II. Non-Standard Form, but can be made standard by multiplying ''negative(s)''.

Example: Solve (-x-2)(3x+1)<0, this is not in standard form, but we can multiply (-x-2) by -1, which yields, (x+2)(3x+1)>0. So the solution will be tex2html_wrap_inline47

Type III. Can't be made standard, needs to use a table

Example: Solve (3x+1)2(x-2)(x-3)<0. First we get tex2html_wrap_inline51 and 3. Next we complete the following table: tex2html_wrap_inline55 Therefore the answer is (2,3).

Example: Solve tex2html_wrap_inline59 According the table above, the answer will be tex2html_wrap_inline61 or tex2html_wrap_inline63

Fractional Inequalities

Example: Solve tex2html_wrap_inline65 This is equivalent to solve (x-1)(x-3)(x-5)>0. We know that we can apply the ''standard form'' technique to do this one.

Example: Solve tex2html_wrap_inline69 This is equivalent to solve (x-1)2(x-3)(x-5)>0 and we can make a table to solve this problem.