Overview:
Number sentences involving absolute value refer to a number’s distance from zero. Absolute value has a number of applications, including measurements and tolerances. It is important to read problems carefully that involve absolute value.
What is Absolute Value?
The mathematical definition of absolute value is a number’s distance from zero on the number line. It is always a positive number, as the distance from 0 for the number 4 and the number -4 is the same. Multiplying the absolute value of |ab| is the same as |a| ∙ |b|.
How To Solve Problems Using the Number Line
A sentence such as |x-3| = 6 can be solved by graphing it on the number line. What the problem means in that the distance between x and 3 is 6 units in either direction. Therefore, one can go forward 6 units from 3 to 9, or backward 6 units, past 0 to -3. The solution set is {-3, 9}, because there are always two values that solve the number sentence, a positive one and a negative one.
How to Solve Problems Using a Compound Sentence
The sentence |x-3| = 6 actually means two different things, either x -3 = 6 or x-3 = -6. This is also a way to check to make sure the solution set is correct. The sentence x-3 = 6 is solved by the positive 9 and the sentence x -3 = -6 is solved by the negative 3.
Some Applications of Absolute Value
Absolute value can be applied when the direction of a difference does not matter, just that there is a difference. For example, many parts are manufactured with a tolerance of measurement. For example, a shaft might be measured with a diameter of 10 mm ± .04 mm. That tolerance could be measured by the sentence |x-10| = .04. As in the above example, there are two sentences within the compound sentence, either x – 10 = .04 or x – 10 = -.04. The shaft could measure as small as 9.96 mm (because 9.96 – 10 = -.04 mm) or as large as 10.04 mm (because 10.04 – 10 = .04 mm).
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