Textbook Question
Use set notation, and list all the elements of each set. {12, 13, 14,..., 20}
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Ch. R - Review of Basic Concepts
Lial13th EditionCollege AlgebraISBN: 9780136881063
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Table of contents
Chapter 1, Problem 20
Determine whether each statement is true or false. |8-12| = |8| - |12|
Verified step by step guidance1
Recall the definition of absolute value: for any real number \(x\), \(|x|\) represents the distance of \(x\) from zero on the number line, and it is always non-negative.
Evaluate the left side of the equation: \(|8 - 12|\). First, calculate the expression inside the absolute value: \(8 - 12 = -4\). Then, find the absolute value: \(|-4|\).
Evaluate the right side of the equation: \(|8| - |12|\). Calculate each absolute value separately: \(|8|\) and \(|12|\), then subtract the results.
Compare the two results from the left and right sides to determine if the equation \(|8 - 12| = |8| - |12|\) holds true.
Remember that the absolute value of a difference is not generally equal to the difference of absolute values, so consider this property when making your conclusion.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Absolute Value Definition
The absolute value of a number represents its distance from zero on the number line, always yielding a non-negative result. For example, |8| = 8 and |-12| = 12, regardless of the sign of the original number.
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Vertex Form
Properties of Absolute Value
Absolute value has specific properties, such as |a - b| representing the distance between numbers a and b. However, absolute value does not distribute over subtraction, meaning |a - b| is not generally equal to |a| - |b|.
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Evaluating True or False Statements
To determine if a statement involving absolute values is true or false, substitute values and apply definitions and properties. For example, calculate |8 - 12| and compare it to |8| - |12| to verify the equality.
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Related Practice
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Write each mixed number as an improper fraction.
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Use specific values for x and y to show that, in general, 1/x + 1/y is not equivalent to 1 / x + y.
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