Textbook Question
Write each English sentence as an equation in two variables. Then graph the equation. The y-value is two more than the square of the x-value.
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 50
Exercises 41–60 contain rational equations with variables in denominators. For each equation, a. write the value or values of the variable that make a denominator zero. These are the restrictions on the variable. b. Keeping the restrictions in mind, solve the equation. 3/(x + 4) - 7 = - 4/(x + 4)
Verified step by step guidance1
Identify the denominators in the equation: here, the denominators are both \( x + 4 \).
Find the values of \( x \) that make the denominator zero by solving \( x + 4 = 0 \). This gives the restriction \( x \neq -4 \) because division by zero is undefined.
Rewrite the equation \( \frac{3}{x + 4} - 7 = -\frac{4}{x + 4} \) and aim to isolate the variable by eliminating the denominators. Since both fractions have the same denominator, consider multiplying both sides of the equation by \( x + 4 \) to clear the denominators, keeping in mind the restriction \( x \neq -4 \).
After multiplying through by \( x + 4 \), simplify the resulting equation by combining like terms and isolating \( x \) on one side.
Solve the simplified equation for \( x \), then check your solution against the restriction \( x \neq -4 \) to ensure it is valid.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Restrictions on the Variable
When solving rational equations, identify values that make any denominator zero, as these are undefined and must be excluded from the solution set. For example, if the denominator is (x + 4), then x = -4 is a restriction because it makes the denominator zero.
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Equations with Two Variables
Solving Rational Equations
To solve rational equations, first eliminate denominators by multiplying both sides by the least common denominator (LCD). This transforms the equation into a simpler form without fractions, making it easier to solve for the variable.
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Checking Solutions Against Restrictions
After solving the equation, substitute the solutions back into the original denominators to ensure none violate the restrictions. Any solution that makes a denominator zero must be discarded as extraneous.
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Related Practice
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