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# How to solve parameters

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Examples with parameters are a special kind of mathematical problems, requiring an off-the-shelf approach in the solution.

Instructions

1

With parameters there can be both equations and inequalities. In either case, we need to express x.

Just in this type of examples, this will be done not explicitly, but through this very parameter.
The parameter itself, or rather its value, isnumber. Usually, the parameters are denoted by the letter a. But the problem is that we do not know its module or sign. This makes it difficult to work with inequalities or the disclosure of modules.

2

Nevertheless, it is possible (but carefully, having previously noted all possible restrictions) to apply all the usual methods of working with equations and inequalities.
And, in principle, the very expression of x through a usually does not take much time and effort.
But writing a full answer is a much more painstaking and time-consuming process.

3

The matter is that in connection with ignorance of the parameter value, we are obliged to consider all possible cases for all values ​​of a from minus to plus infinity.
Here we will need a graphical method. Sometimes it is also called a "coloring". It consists in the fact that we represent the lines obtained in the transformation of our original example in the x (a) (or a (x) - as convenient) axes. And then we start to work with these lines: since the value of a is not fixed, we need the lines containing the parameter in the equation to shift along the graph, simultaneously tracing and calculating the intersection points with other lines, and also analyzing the signs of the regions: they fit us or no. Suitable for convenience and visibility we shaded.
Thus, we go through the entire numerical axis from minus to plus infinity, checking the answer for all a.

4

The answer itself is written similarly to the answer forMethod of intervals with some proviso: we do not simply indicate the set of solutions for x, but write to which set of values ​​of a there corresponds a set of values ​​of x.