Absolute Value
Absolute value represents the distance of a real number from zero on the number line, always yielding a positive result. It is denoted algebraically as |x|, pronounced "the absolute value of x." For instance, the absolute values of 5 and -12 are both 5 and 12, respectively. This mathematical concept is particularly useful in problem-solving scenarios where distances or differences are involved, allowing for a consistent positive result regardless of the direction of the numbers involved.
When comparing numbers, one should first take their absolute values; for example, the comparison of |-7| and |-5| reveals that 7 is greater than 5, even though -7 is less than -5. In inequalities, absolute values can lead to multiple solutions, expressed using "and" when rewriting the inequality to account for both positive and negative scenarios. Conversely, equations involving absolute values often yield two solutions, requiring the use of "or" to present the distinct answers. For example, the equation |x - 6| = 12 leads to two possible values for x. Understanding these principles can greatly enhance one’s ability to tackle mathematical problems involving distances and comparisons.
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Absolute Value
The absolute value of real numbers can be found by finding how far that number is from zero. This distance it is always positive. Mathematicians write absolute value algebraically, |x| which is read "the absolute value of x." Any real number located within the || is the distance from zero, to be determined. Here are a couple of examples |5| and |–12|. The answer for each example is 5 and 12 respectively.
Constructing and interpreting absolute value is a valuable tool. When you are looking at a problem that can be viewed from different aspects, absolute value can be used to solve the problem. Here is an example: A castle sits near a moat which begins B meters from the castle and ends E meters from the castle. How wide is the moat? In this example, either side of the moat can be the beginning or the end you could come up with B – E or E – B. Due to this fact, utilizing absolute value ensures we get a positive value for our answer, therefore to answer the given example |B – E| and |E – B| will produce the correct answer.
When comparing absolute values it is important to remember to start by taking the absolute value first. Once the absolute value is taken of the number, the comparison to the number can be done. Take |–7| and | – 5| for instance. If you compare the real numbers within the ||, –7 is less than –5, which is a true statement, however it is not the answer to the question. After the absolute value of each number is taken, the comparison gives 7 is greater than 5, which is the answer to the question.
Inequalities include, in most instances, multiple answers so utilize the "and" when solving for the answers. Finding absolute values in inequalities starts with a two-step process of rewriting the inequality with both a positive and a negative answer. This is due to the fact that absolute value always gives a positive answer regardless of whether or not the answer is positive. Given |x| < 12; rewrite as x > –12 and x < 12; which can be written as –12 < x < 12. This indicates that all numbers that fall between –12 and +12 make the inequality true. Given |3x + 5| < 7, rewrite as: 3x + 5 > –7 and 3x + 5 < 7 and then solve each: so x > –4 and x < 2/3; which can be written as –4 < x < 2/3. Similarly any numbers that fall between –4 and 2/3 make this inequality true. Unlike inequalities, equations that include absolute values have two distinct answers and therefore utilize the term "or" instead of "and." Given |x – 6| = 12 you would again rewrite as two equations, however use "or" between the two equations: x – 6 = 12 or x – 6 = –12, and solve. For this example x = 18 or x = –6 to make the equation true.
Bibliography
Fosnot, Catherine T., and Bill Jacob. Young Mathematicians at Work: Constructing Algebra. Portsmouth, Heinemann, 2010.
Greenes, Carole E., and Rheta Rubenstein, eds. Algebra and Algebraic Thinking in School Mathematics. National Council of Teachers of Mathematics 70th Yearbook. Reston: NCTM, 2008.
Van de Walle, John, Karen S. Karp, and Jennifer Bay-Williams. Elementary and Middle School Mathematics Methods: Teaching Developmentally. New York: Allyn, 2010.