Textbook Question
In Exercises 59–94, solve each absolute value inequality. |(2x + 2)/4| ≥ 2
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Ch. 1 - Equations and Inequalities
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 2, Problem 74
Exercises 73–75 will help you prepare for the material covered in the next section. Simplify: √18 - √8
Verified step by step guidance1
Identify the radicals to simplify: \( \sqrt{18} - \sqrt{8} \).
Express each radicand as a product of a perfect square and another factor: \( 18 = 9 \times 2 \) and \( 8 = 4 \times 2 \).
Rewrite each square root using the property \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \): \( \sqrt{18} = \sqrt{9} \times \sqrt{2} \) and \( \sqrt{8} = \sqrt{4} \times \sqrt{2} \).
Simplify the square roots of the perfect squares: \( \sqrt{9} = 3 \) and \( \sqrt{4} = 2 \), so the expression becomes \( 3\sqrt{2} - 2\sqrt{2} \).
Combine like terms by subtracting the coefficients of \( \sqrt{2} \): \( (3 - 2)\sqrt{2} \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Simplifying Square Roots
Simplifying square roots involves expressing the radicand as a product of perfect squares and other factors. For example, √18 can be rewritten as √(9×2), which simplifies to 3√2 because √9 is 3. This process makes it easier to combine or subtract roots.
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Like Radicals
Like radicals have the same radicand and index, allowing them to be combined through addition or subtraction. For instance, 3√2 and 5√2 are like radicals and can be combined as (3 - 5)√2 = -2√2. Recognizing like radicals is essential for simplifying expressions involving roots.
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Subtraction of Radicals
To subtract radicals, first simplify each radical and then combine like radicals by subtracting their coefficients. If the radicals are not like terms, they cannot be combined further. For example, √18 - √8 simplifies to 3√2 - 2√2, which equals √2.
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Related Practice
Textbook Question
Solve each equation in Exercises 65–74 using the quadratic formula. x2 - 6x + 10 = 0
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Textbook Question
Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The ordered pair (2, 5) satisfies 3y - 2x = - 4.
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Textbook Question
Solve each equation by the method of your choice.
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Textbook Question
In Exercises 71–78, solve each equation. Then determine whether the equation is an identity, a conditional equation, or an inconsistent equation. 4(x + 5) = 21 + 4x
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Textbook Question
Exercises 73–75 will help you prepare for the material covered in the next section. Multiply: (7 - 3x)(- 2 - 5x)
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