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Chapter 6: Problem 8

Give the domain of the variable in each equation. $$\frac{3}{x+4}-\frac{2}{x-9}=0$$

Short Answer

Expert verified
The domain is all real numbers except \(x = -4\) and \(x = 9\).

Step by step solution

01

- Identify Restrictions

The variable in this equation is in the denominators. The domain consists of all real numbers except those which make any denominator zero. Identify where the denominators equal zero: 1. Determine when \(x + 4 = 0\).2. Determine when \(x - 9 = 0\).
02

- Solve for Restrictions

Solve the equations obtained from Step 1 to find the restrictions.For \(x + 4 = 0\): \(x = -4\).For \(x - 9 = 0\): \(x = 9\).
03

- State the Domain

The domain is the set of all real numbers except those found in Step 2. Therefore, the domain is all real numbers except \(x = -4\) and \(x = 9\).In interval notation, the domain is: \((-\backslash\backslash\text{infty}, -4) \backslashcup (-4, 9) \backslashcup (9, \backslash\backslash\text{infty})\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

rational expressions
A rational expression is a fraction where both the numerator and the denominator are polynomials.
In simple terms, it's like an improved version of a fraction where we have polynomials instead of numbers. For example, in our problem, the expression \( \frac{3}{x+4} - \frac{2}{x-9} \) is a rational expression.

Understanding rational expressions is crucial because:
  • They help us understand more complex algebraic concepts.
  • Knowing how to handle them will make solving equations easier.
  • They appear frequently in higher-level math, so getting comfortable with them now is beneficial for future studies.
Just remember, whenever you're dealing with rational expressions, always keep an eye on the denominator because it leads us to domain restrictions!
domain restrictions
In mathematics, the domain of a variable is the set of all possible values that the variable can take. However, when dealing with rational expressions, we must be careful about domain restrictions.
Domain restrictions occur when the denominator of a fraction equals zero because dividing by zero is undefined.

Let's look at our specific problem:
  • The denominators are \(x+4\) and \(x-9\).
  • To find domain restrictions, we set each denominator to zero:
    \(x+4=0\) leads to \(x=-4\), and
    \(x-9=0\) leads to \(x=9\).
  • So, \(x\) cannot be \(-4\) or \(9\).
Thus, our domain consists of all real numbers except where the denominators are zero. This ensures that our expression remains valid and avoids undefined values.
interval notation
Interval notation is a method to describe sets of numbers, typically to express the domain of a variable in a more compact form.

It uses brackets and parentheses to show which numbers are included or excluded:
  • \([ ... ]\) : includes the endpoints
  • \( ( ... )\) : excludes the endpoints
  • \cup\ \: union symbol, combines intervals
For our example, the domain in interval notation is \( (-\infty, -4) \cup (-4, 9) \cup (9, \infty) \).
This means:
  • First interval: all numbers less than \(-4\)
  • Second interval: all numbers between \(-4\) and \(9\)
  • Third interval: all numbers greater than \(9\)
Using interval notation provides clarity and precision when describing sets of numbers, especially for domain and range in algebra and calculus.

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Most popular questions from this chapter

As review, multiply or divide the rational numbers as indicated. Write answers in lowest terms. $$ \frac{3}{8} \div \frac{5}{12} $$

Add or subtract as indicated. $$\frac{3}{(p-2)^{2}}-\frac{5}{p-2}+4$$

Multiply or divide as indicated. $$ \frac{8 a^{2}-6 a-9}{6 a^{2}-5 a-6} \div \frac{4 a^{2}+11 a+6}{9 a^{2}+12 a+4} $$

Write each rational expression in lowest terms. $$ \frac{y^{3}(y-4)}{y^{2}(y-4)} $$

Give the domain of each rational function using (a) set-builder notation and (b) interval notation. $$ f(x)=\frac{9 x^{2}-8 x+3}{4 x^{2}+1} $$
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