Textbook Question
Determine whether each statement makes sense or does not make sense, and explain your reasoning. I used the ordered pairs (- 2, 2), (0, 0), and (2, 2) to graph a straight line.
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 70a
Find all values of x such that y = 0. y = 1/(5x + 5) - 3/(x + 1) + 7/5
Verified step by step guidance1
Rewrite the equation y = 0 as 1/(5x + 5) - 3/(x + 1) + 7/5 = 0. The goal is to solve for x.
Identify the least common denominator (LCD) of the fractions. The denominators are (5x + 5), (x + 1), and 5. Factorize (5x + 5) as 5(x + 1), so the LCD is 5(x + 1).
Multiply through the entire equation by the LCD, 5(x + 1), to eliminate the fractions. This gives: 5(x + 1) * [1/(5x + 5)] - 5(x + 1) * [3/(x + 1)] + 5(x + 1) * [7/5] = 0.
Simplify each term after multiplying by the LCD. For the first term, the (5x + 5) cancels, leaving 1/(5x + 5) * 5(x + 1) = (x + 1). For the second term, the (x + 1) cancels, leaving -3. For the third term, the 5 cancels, leaving 7(x + 1). Combine these to form a simplified equation: (x + 1) - 3 + 7(x + 1) = 0.
Combine like terms in the simplified equation. Expand and simplify to form a linear equation in terms of x. Solve for x by isolating it on one side of the equation.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Rational Functions
A rational function is a function that can be expressed as the ratio of two polynomials. In this case, the function involves terms like 1/(5x + 5) and -3/(x + 1), which are rational expressions. Understanding how to manipulate and simplify these expressions is crucial for solving equations involving them.
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Finding Roots of an Equation
Finding the values of x such that y = 0 involves solving the equation for its roots. This means determining the x-values where the function intersects the x-axis. This process often requires setting the entire expression equal to zero and solving for x, which may involve combining like terms and factoring.
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Domain Restrictions
When dealing with rational functions, it is important to consider the domain, which includes all possible x-values for which the function is defined. In this case, values that make the denominators zero (like x = -1 and x = -1) must be excluded from the solution set, as they lead to undefined expressions.
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Related Practice
Textbook Question
In Exercises 67–70, find all values of x such that y = 0.
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Textbook Question
In Exercises 61–76, solve each absolute value equation or indicate that the equation has no solution. 2|4 - (5/2)x| + 6 = 18
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Textbook Question
Evaluate x2 - 2x + 2 for x = 1 + i.
Textbook Question
Solve each equation using the quadratic formula.
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Textbook Question
Solve each equation in Exercises 65–74 using the quadratic formula.