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Ch. R - Algebra Review
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook
All textbooksLial 12th EditionCh. R - Algebra ReviewProblem 34
Chapter 1, Problem 34

Let A = {-6, -12⁄4, -5⁄8, -√3, 0, ¼, 1, 2π, 3, √12}. List all the elements of A that belong to each set. Real numbers

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1
Recall that the set of real numbers includes all rational and irrational numbers, including integers, fractions, and irrational roots, but excludes complex numbers with imaginary parts.
Examine each element of the set \(A = \{-6, -\frac{12}{4}, -\frac{5}{8}, -\sqrt{3}, 0, \frac{1}{4}, 1, 2\pi, 3, \sqrt{12}\}\) to determine if it is a real number.
Note that \(-6\) is an integer, so it is a real number.
Simplify \(-\frac{12}{4}\) to \(-3\), which is an integer and thus a real number.
Recognize that \(-\frac{5}{8}\) is a rational number (a fraction), so it is real.
Understand that \(-\sqrt{3}\) is an irrational number (since \(\sqrt{3}\) is irrational), but still a real number.
Note that \(0\) is a real number.
Recognize that \(\frac{1}{4}\) is a rational number, so it is real.
Note that \(1\) is an integer and thus real.
Understand that \(2\pi\) is a real number because \(\pi\) is irrational but real, and multiplying by 2 keeps it real.
Note that \(3\) is an integer and real.
Simplify \(\sqrt{12}\) to \(2\sqrt{3}\), which is irrational but real.
Conclude that all elements in set \(A\) are real numbers.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Real Numbers

Real numbers include all rational and irrational numbers, encompassing integers, fractions, decimals, and roots. They can be represented on the number line and do not include imaginary or complex numbers. Understanding this set helps identify which elements from a given list are real.
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Introduction to Complex Numbers

Rational and Irrational Numbers

Rational numbers can be expressed as a fraction of two integers, while irrational numbers cannot be written as simple fractions and have non-repeating, non-terminating decimals. Recognizing these helps classify elements like -5/8 (rational) and -√3 (irrational) within the real numbers.
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Rationalizing Denominators

Simplification of Expressions

Simplifying expressions such as fractions and roots is essential to accurately identify and compare numbers. For example, -12/4 simplifies to -3, and √12 simplifies to 2√3, aiding in clearer classification within sets like real numbers.
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Simplifying Trig Expressions
Related Practice
Textbook Question

Multiply or divide, as indicated. See Example 3. ((2k + 8) / 6) ÷ ((3k + 12) / 2)

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Textbook Question

Concept Check Plot each point, and then plot the points that are symmetric to the given point with point with respect to the (c) origin. (5, -3)

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Textbook Question

Determine whether each relation defines y as a function of x. Give the domain and range. See Example 5. y = 2x - 5

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Textbook Question

Concept Check Plot each point, and then plot the points that are symmetric to the given point with point with respect to the (b) y-axis (5, -3)

Textbook Question

For each equation, (a) give a table with at least three ordered pairs that are solutions, and (b) graph the equation. See Examples 3 and 4.

y = √(x - 3)