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# Module 18

### Nonlinear inequalities in One Variable

The first step to solve a quadratic inequality is to change the greater than, less than, greater than or equal to sign, or less than or equal to sign to what sign? The equal sign
Setting each factor of the quadratic inequality to zero and solving is taking advantage of what property? The zero factory property : Ex: Allows you to turn (X+3)(x-2) = 0 Into x=-3 and x=2 to solve for x
After setting each equation to zero and using the zero factor property to solve, each equation will give you an integer. What is needed next so find solution sets for the inequality? A number line separated into regions with the solutions
After solving for variables you will plot the solutions on the number line. What is the next step? Checking a number from each proposed solution region in the original inequality to see if a true statement results.
Solution sets are noted with interval notation. Special rules apply. For greater than or less than signs, parentheses are used. For greater than or equal to, or less than or equal to brackets are used.
To solve a rational inequality what are the steps? 1. Solve for values that make all denominators zero. 2. Solve the related equation. 3. Separate the number line into regions with the solutions you obtained.
When working with rational inequalities the solution set does not include what? Any values that would make any denominator zero.
Practice Problem: x+3 ------- ≥ 0 x-7 After solving you would get that x is ≤-3, pr ≥ 7. What error would this cause? If you include 7 in the solution set the resulting denominator would be zero, which you can not divide by. Instead we will notate this as x>7, so 7 is not included. The solution set becomes (-infinity,-3]U(7,infinity)
To make the numerator solvable in a rational inequality what must be done? Multiply the entire problem by the entire denominator term. This will remove the denominator and leave you with a inequality that can be solved using the zero factor property. Which will provide you with another point to plot on a number line.
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