Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Concept explainers
Equations and Inequations
Equations and inequalities describe the relationship between two mathematical expressions.
Linear Functions
A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.
Question
![### Solving Rational Equations
**Problem Statement:**
Solve for \(x\):
\[
\frac{x}{x + 4} + \frac{5}{x - 7} = \frac{-9x + 8}{x^2 - 3x - 28}
\]
**Steps to Solve:**
1. **Identify Common Denominator:**
- Factor the quadratic expression in the denominator on the right side.
\[
x^2 - 3x - 28 = (x - 7)(x + 4)
\]
2. **Rewrite the Equation:**
- Using the factorized form, rewrite the equation:
\[
\frac{x}{x + 4} + \frac{5}{x - 7} = \frac{-9x + 8}{(x - 7)(x + 4)}
\]
3. **Combine Fractions on the Left Side:**
- To combine the fractions on the left side, obtain a common denominator:
\[
\frac{x(x - 7) + 5(x + 4)}{(x - 7)(x + 4)} = \frac{-9x + 8}{(x - 7)(x + 4)}
\]
4. **Simplify the Numerators:**
- Expand and simplify the numerators:
\[
\frac{x^2 - 7x + 5x + 20}{(x - 7)(x + 4)} = \frac{-9x + 8}{(x - 7)(x + 4)}
\]
- Simplify further:
\[
\frac{x^2 - 2x + 20}{(x - 7)(x + 4)} = \frac{-9x + 8}{(x - 7)(x + 4)}
\]
5. **Equate the Numerators:**
- Since the denominators are equal, set the numerators equal to each other:
\[
x^2 - 2x + 20 = -9x + 8
\]
6. **Solve the Quadratic Equation:**
- Rearrange the equation to the standard quadratic form:
\[
x^2 + 7x + 12 = 0](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5f072395-e0a4-486f-9c92-023b53ab9ed9%2Fd8a57973-57d9-4029-aa91-304702257d8e%2Fednlunc_processed.png&w=3840&q=75)
Transcribed Image Text:### Solving Rational Equations
**Problem Statement:**
Solve for \(x\):
\[
\frac{x}{x + 4} + \frac{5}{x - 7} = \frac{-9x + 8}{x^2 - 3x - 28}
\]
**Steps to Solve:**
1. **Identify Common Denominator:**
- Factor the quadratic expression in the denominator on the right side.
\[
x^2 - 3x - 28 = (x - 7)(x + 4)
\]
2. **Rewrite the Equation:**
- Using the factorized form, rewrite the equation:
\[
\frac{x}{x + 4} + \frac{5}{x - 7} = \frac{-9x + 8}{(x - 7)(x + 4)}
\]
3. **Combine Fractions on the Left Side:**
- To combine the fractions on the left side, obtain a common denominator:
\[
\frac{x(x - 7) + 5(x + 4)}{(x - 7)(x + 4)} = \frac{-9x + 8}{(x - 7)(x + 4)}
\]
4. **Simplify the Numerators:**
- Expand and simplify the numerators:
\[
\frac{x^2 - 7x + 5x + 20}{(x - 7)(x + 4)} = \frac{-9x + 8}{(x - 7)(x + 4)}
\]
- Simplify further:
\[
\frac{x^2 - 2x + 20}{(x - 7)(x + 4)} = \frac{-9x + 8}{(x - 7)(x + 4)}
\]
5. **Equate the Numerators:**
- Since the denominators are equal, set the numerators equal to each other:
\[
x^2 - 2x + 20 = -9x + 8
\]
6. **Solve the Quadratic Equation:**
- Rearrange the equation to the standard quadratic form:
\[
x^2 + 7x + 12 = 0
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