Textbook Question
Perform the indicated operations and write the result in standard form. (1 + i)/(1 + 2i) + (1 - i)/(1 - 2i)
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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 96a
Use interval notation to represent all values of x satisfying the given conditions. y1 = (2/3)(6x - 9) + 4, y2 = 5x + 1, and y1 > y2
Verified step by step guidance1
Step 1: Start by setting up the inequality y1 > y2. Substitute the given expressions for y1 and y2 into the inequality: \( \frac{2}{3}(6x - 9) + 4 > 5x + 1 \).
Step 2: Simplify the left-hand side of the inequality. Distribute \( \frac{2}{3} \) across \( (6x - 9) \): \( \frac{2}{3} \cdot 6x - \frac{2}{3} \cdot 9 + 4 > 5x + 1 \). This simplifies to \( 4x - 6 + 4 > 5x + 1 \).
Step 3: Combine like terms on the left-hand side: \( 4x - 2 > 5x + 1 \).
Step 4: Isolate x by subtracting \( 4x \) from both sides: \( -2 > x + 1 \). Then subtract 1 from both sides: \( -3 > x \), or equivalently \( x < -3 \).
Step 5: Represent the solution in interval notation. Since \( x < -3 \), the interval is \( (-\infty, -3) \).
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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 mathematical notation used to represent a range of values on the number line. It uses parentheses and brackets to indicate whether endpoints are included (closed intervals) or excluded (open intervals). For example, the interval (2, 5] includes all numbers greater than 2 and up to 5, including 5 but not 2.
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Interval Notation
Inequalities
Inequalities express the relationship between two expressions that are not necessarily equal. In this context, the inequality y1 > y2 indicates that the values of y1 must be greater than those of y2 for certain values of x. Understanding how to manipulate and solve inequalities is crucial for determining the range of x that satisfies the given condition.
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Linear Functions
Linear functions are mathematical expressions that create a straight line when graphed. They can be represented in the form y = mx + b, where m is the slope and b is the y-intercept. In this problem, both y1 and y2 are linear functions of x, and analyzing their intersection points will help determine the values of x that satisfy the inequality.
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Related Practice
Textbook Question
Solve each equation in Exercises 96–102 by the method of your choice. 2√(x-1) = x
Textbook Question
In Exercises 91–100, find all values of x satisfying the given conditions. and
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Textbook Question
Solve each equation in Exercises 83–108 by the method of your choice.
Textbook Question
Solve each equation in Exercises 83–108 by the method of your choice.
Textbook Question
Perform the indicated operations and write the result in standard form. 8/(1 + 2/i)