PDF Advanced inequalities

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They deal with relations of variables denoted by four signs:. For two numbers and :. Note that if and only if , , and vice versa. The same applies to the latter two signs: if and only if , , and vice versa. A common application of inequalities is solving them for a variable.

For example, consider the inequality. We can solve for the variable here and get , thus placing implicit restrictions upon the variable.


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A more complex example is. Here is a common mistake:. The problem here is that we multiplied by as one of the last steps.


We also kept the inequality sign in the same direction. However, we don't know if the quantity is negative or not; we can't assume that it is positive for all real. Thus, we may have to reverse the direction of the inequality sign if we are multiplying by a negative number.

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But, we don't know if the quantity is negative either. A correct solution would be to move everything to the left side of the inequality, and form a common denominator. Then, it will be simple to find the solutions to the inequality by considering the sign negativeness or positiveness of the fraction as varies:. We will start with an intuitive solution, and then a rule can be built for solving general fractional inequalities. To make things easier, we test positive integers. Nor does. Therefore, these two aren't solutions. Then we begin to test numbers such as , , and so on. All of these work.

In fact, it's not difficult to see that the fraction will remain positive as gets larger and larger. But just where does , which causes a negative fraction at and , begin to cause a positive fraction? We can't just assume that is the switching point; this solution is not simply limited to integers. The numerator and denominator are big hints. Specifically, we examine that when the numerator , then the fraction is , and begins to be positive for all higher values of. Solving the equation reveals that is the turning point.

After more of this type of work, we realize that brings about division by , so it certainly isn't a solution.

Also included in:. Save money by buying the bundle and be prepared for an entire unit of engaging activities! View Bundle. One Step Inequalities Bundle. With this bundle you get my 6 One Step Inequalities Activities. Save money by buying the bundle and be prepared for an entire unit of engaging one step inequality activities! Included are 5 Mazes and 1 Card Sort Activity.

High School Math: Advanced Math and Precalculus / Systems of Inequalities

The mazes are a great way to teach the different types of one step inequalities. Important InformationDue to how TPT auto-creates bundles this will download as one un-organized folder of resources. Simply open the bonus file to access a version that's already been. Simply open the bonus file to access a version that's already been sorted into folders f.

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Advanced Algebra: Algebraic Inequalities – II

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