Problem 1 - College Algebra and Calculus: An Applied Approach

Question

In Exercises \(1-4\), determine if each value of \(x\) is in the domain of the expression.\(\frac{x+2}{5 x+2}\)

Step-by-Step Solution

Answer

The domain of the function \(\frac{x+2}{5x+2}\) is all real numbers except \(x = -\frac{2}{5}\).

Step 1: Identify the denominator of the expression

The denominator of our expression is \(5x + 2\). The denominator cannot equal zero as division by zero is undefined.

Step 2: Solve the denominator equation for zero

Set the denominator equal to zero and solve for \(x\). So, the equation is \(5x + 2 = 0\). To solve this, isolate \(x\). First, subtract 2 from both sides of the equation to have \(5x = -2\). Then divide by 5 to get \(x = -\frac{2}{5}\).

Step 3: Determine the domain of the function

The domain of the function \(\frac{x+2}{5x+2}\) is all real numbers except for \(x = -\frac{2}{5}\), as this value makes the denominator zero.

Key Concepts

Understanding Division by Zero

Division by zero is a fundamental concept in mathematics that often leads to confusion. When a number is divided by zero, the operation is undefined. This is because there is no number that can multiply with zero to give any number other than zero itself.

When dealing with functions or rational expressions, the denominator cannot be zero. If the denominator turns zero, the function doesn't exist for that value of x. Consider the expression \( \frac{x+2}{5x+2} \). The task is to find where the denominator, \( 5x+2 \), equals zero. Thus, it's crucial to solve for \( x \) when setting the denominator to zero. This calculation prevents division by zero and helps understand the conditions under which the function is defined.

Always remember:

• A function is undefined where the denominator is zero.
• Set the denominator equal to zero and solve for \( x \) to find excluded values.
• Excluding these \( x \)-values gives the domain of the function.

Getting to Know Rational Expressions

Rational expressions are fractions that have polynomials in the numerator and the denominator. An example is \( \frac{x+2}{5x+2} \). These expressions are crucial in algebra and help describe various relationships and functions.

To analyze rational expressions, first identify the denominator, as its value determines the expression's domain. Remember, the denominator dictates when the expression is undefined (as we discussed with division by zero).

When simplifying rational expressions:

• Factor both the numerator and the denominator.
• Cancel common factors, if possible.
• Ensure to note any restrictions on the variable, often caused by possible division by zero.

By understanding the properties of rational expressions, you can solve equations, simplify expressions, and evaluate limits more effectively and accurately.

The Art of Solving Equations

Solving equations is a basic skill in mathematics, where the goal is to find all values of \( x \) that make the equation true. For rational expressions, it's slightly more complex, as you deal with the possibility of division by zero.

Here's a step-by-step guide to solving an equation like \( 5x + 2 = 0 \):

• Isolate the variable term by performing inverse operations. For \( 5x + 2 = 0 \), subtract 2 from both sides to get \( 5x = -2 \).
• Solve for \( x \) by dividing both sides by 5. This results in \( x = -\frac{2}{5} \).
• Check the solution to ensure it does not cause division by zero in the original expression.

By following these steps, you ensure that you not only find potential solutions but also verify that they are valid within the context of the problem. Such practices solidify the foundation of solving various types of equations in mathematics.