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Solving Absolute Value Inequalities (AVI) Step-by-Step

What is the best method for solving absolute value inequalities?
The first most important part about solving absolute value inequalities is to understand that there are two types of inequalities. These are |ax+b|<c and |ax+b|>c. These inequalities have totally different solutions. This is because of the difference between the two basic inequalities |x|<c and |x|>c. Absolute value is distance from 0. If |x|<c this means that the distance between x and 0 must be less than c. So then x must be between -c and c and we get a single interval (-c,c). If |x|>c, then the distance between x and 0 is greater than c. So x is either far away from 0 to the left or far away from 0 to the right. Now there are 2 intervals: (-infinity,-c) ; (c,infinity). We show how to "see" and solve these 2 types step-by-step.
Finally, we remember that |x|=|-x|. This means that we do not solve absolute value inequalities with a negative x inside! We simply rewrite it: |2-3x|=|3x-2|. This means we never have to worry about dividing by a negative number.

Solution Preview

Absolute Value Inequalities

The most important part about solving absolute value inequalities is to understand that there are two types of inequalities: "less than" and "greater than" (they may include equals).
The two types look like |ax+b|<c or |ax+b|>c where a,b and c are (positive or negative) numbers.

The second important part to solving is to see the difference between the two basic inequalities |x|<c and |x|>c using the number line. The absolute value of x is the distance from x to 0.

If |x|<c, this means that the distance from x to 0 must be less than c. So then x must be between -c and c. So the solution is the single piece of number line from -c to c. This is the interval ...

Solution Summary

Here we show how to solve the two types of absolute value inequalities |ax+b|<c and |ax+b|>c. We start by explaining and then solving the two basic inequalities |x|<c and |x|>c using the number line. We then use this solution to solve the mid-level inequalities |x+b|<c and |x+b|>c. We then use this solution to solve the high-level inequalities |ax+b|<c and |ax+b|>c. In this way both the meaning and solution method is clear and step-by-step.

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