Ch. 1 - Equations and Inequalities

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

Lial 13th Edition

Ch. 1 - Equations and Inequalities

Problem 12a

Chapter 2, Problem 12a

Identify each number as real, complex, pure imaginary, or nonreal com-plex. (More than one of these descriptions will apply.) 0

Verified step by step guidance

Recall the definitions: A real number is any number that can be found on the number line, including zero, positive, and negative numbers.

A complex number is any number that can be written in the form $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit with $i^2 = -1$.

A pure imaginary number is a complex number where the real part $a = 0$ and the imaginary part $b \neq 0$.

A nonreal complex number is a complex number where the imaginary part $b \neq 0$, meaning it is not purely real.

Since the number given is $0$, it can be written as \$0 + 0i$, which means it is a real number and also a complex number, but it is neither pure imaginary nor nonreal complex.

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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 the numbers that can be found on the number line, including zero, positive and negative integers, fractions, and irrational numbers. They do not have an imaginary component and are used to measure continuous quantities.

Complex Numbers

Complex numbers consist of a real part and an imaginary part and are expressed in the form a + bi, where 'a' and 'b' are real numbers and 'i' is the imaginary unit with the property i² = -1. All real numbers are also complex numbers with zero imaginary part.

Pure Imaginary Numbers

Pure imaginary numbers are complex numbers where the real part is zero and the imaginary part is nonzero, written as 0 + bi (b ≠ 0). They lie on the imaginary axis of the complex plane and are distinct from real numbers.

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Use Choices A–D to answer each question. A. 3x² - 17x - 6 = 0 B. (2x + 5)² = 7 C. x² + x = 12 D. (3x - 1)(x - 7) = 0 Only one of the equations is set up so that the values of a, b, and c can be determined immediately. Which one is it? Solve it.

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Textbook Question

Solve each equation using the zero-factor property. x² - 5x + 6 = 0

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Use the following facts. If x represents an integer, then x+1 represents the next consecutive integer. If x represents an even integer, then x+2 represents the next consecutive even integer. If x represents an odd integer, then x+2 represents the next consecutive odd integer. Find two consecutive odd integers whose product is 63.

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Textbook Question

Solve each problem. See Example 1. Michael must build a rectangular storage shed. He wants the length to be 6 ft greater than the width, and the perimeter will be 44 ft. Find the length and the width of the shed.

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Determine the values of the variable that cannot possibly be solutions of each equation. Do not solve. 3/(x-2) + 1/(x+1) = 3/(x²-x-2)