Textbook Question
Solve each equation in Exercises 15–34 by the square root property.
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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 23
In Exercises 15–26, use graphs to find each set. [3, ∞) ∩ (6, ∞)
Verified step by step guidance1
Identify the two sets given: \([3, \infty)\) and \((6, \infty)\). The first set includes all numbers from 3 to infinity, including 3, and the second set includes all numbers greater than 6, but not including 6.
Visualize each set on a number line: For \([3, \infty)\), shade the line starting at 3 (including 3) and extending to the right indefinitely. For \((6, \infty)\), shade the line starting just after 6 (not including 6) and extending to the right indefinitely.
To find the intersection \([3, \infty) \cap (6, \infty)\), look for the region where both shaded areas overlap on the number line.
Since \((6, \infty)\) starts after 6 and \([3, \infty)\) starts at 3, the overlap will be all numbers greater than 6 (because the second set is more restrictive).
Express the intersection set in interval notation, which will be the set of all numbers greater than 6, not including 6, extending to infinity.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Interval Notation
Interval notation is a way to represent sets of real numbers using brackets and parentheses. Square brackets [ ] include endpoints, while parentheses ( ) exclude them. For example, [3, ∞) includes all numbers from 3 to infinity, including 3.
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05:18
Interval Notation
Set Intersection
The intersection of two sets includes all elements common to both sets. For intervals, this means finding the overlapping region on the number line where both intervals exist simultaneously.
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07:52Parallel & Perpendicular Lines
Graphing Intervals on the Number Line
Graphing intervals involves shading the portion of the number line that corresponds to the interval. This visual helps identify overlaps and intersections by showing where intervals coincide.
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02:35Graphing Lines in Slope-Intercept Form
Related Practice
Textbook Question
Graph each equation in Exercises 13 - 28. Let x = - 3, - 2, - 1, 0, 1, 2, 3
y = -2|x|
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Solve each radical equation in Exercises 11–30. Check all proposed solutions. √(x + 8) - √(x - 4) = 2
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Textbook Question
In Exercises 1–26, solve and check each linear equation. 16 = 3(x - 1) - (x - 7)
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Textbook Question
In Exercises 21–28, divide and express the result in standard form. 2i/(1 + i)
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