Problem 7 - A Graphical Approach to College Algebra

Question

Find the domain of each rational expression. $$\frac{3 x+7}{(4 x+2)(x-1)}$$

Step-by-Step Solution

Answer

The domain is all real numbers except \\( x = -\frac{1}{2} \\\) and \\( x = 1 \\\).

Step 1: Identifying Problematic Denominators

Rational expressions are undefined when their denominator is equal to zero. Thus, finding the domain involves identifying values of the variable that cause the denominator to be zero.

Step 2: Setting Denominator to Zero

To find these values, set the denominator \( (4x + 2)(x - 1) \) equal to zero. This gives us two separate equations: \( 4x + 2 = 0 \) and \( x - 1 = 0 \).

Step 3: Solving First Equation

Solve \( 4x + 2 = 0 \) for \( x \). Subtract 2 from both sides: \( 4x = -2 \). Then divide by 4: \( x = -\frac{1}{2} \).

Step 4: Solving Second Equation

Solve \( x - 1 = 0 \) for \( x \). Add 1 to both sides: \( x = 1 \).

Step 5: Determining the Domain

The domain is all real numbers except for the values that make the denominator zero, which are \( x = -\frac{1}{2} \) and \( x = 1 \). Thus, the domain is \({x \mid x eq -\frac{1}{2}, x eq 1}\).

Key Concepts

Rational Expressions

Rational expressions are mathematical fractions where both the numerator and the denominator are polynomials. They resemble fractions you might be familiar with, but instead of just numbers, they include variables. For example, in the expression \( \frac{3x+7}{(4x+2)(x-1)} \), \( 3x+7 \) is the numerator, and \( (4x+2)(x-1) \) is the denominator.

Working with rational expressions is similar to dealing with regular fractions:

• All the typical fraction rules apply, such as simplifying where possible, multiplying, and adding them.
• Always remember that the denominator should never be zero. This principle is crucial because division by zero is undefined in mathematics.

Understanding this key concept will help when solving problems involving rational expressions, as you will often need to consider when they might become undefined due to their denominators.

Undefined Denominator

In mathematics, a fraction becomes undefined when its denominator equals zero. This is because dividing by zero is not permissible. For rational expressions, it's crucial to determine their domain by identifying any values that make their denominator zero.

For instance, if we consider the rational expression \( \frac{3x+7}{(4x+2)(x-1)} \):

• The denominator \((4x+2)(x-1)\) can be zero at certain values of \(x\).
• Specifically, we set \(4x+2\) and \(x-1\) each equal to zero to find these points.

By detecting these zero points, we can establish the domain of the expression, which includes all real numbers apart from those that make the denominator zero. Thus, the expression is defined for all values of \(x\) except \(x = -\frac{1}{2}\) and \(x = 1\).

Solving Equations

To find the values that make the denominator zero, you need to solve simple equations. This is an essential step in determining the domain of a rational expression.

Let's explore these steps using our previous expression \( \frac{3x+7}{(4x+2)(x-1)} \):

• First, solve the equation \( 4x + 2 = 0 \). Subtract 2 from both sides which gives \( 4x = -2 \). Then divide by 4, resulting in \( x = -\frac{1}{2} \).
• Next, solve \( x - 1 = 0 \). Adding 1 to both sides results in \( x = 1 \).

These solutions tell us precisely where the rational expression is undefined. Once you have the solutions, it becomes straightforward to express the domain, which is every real number except those solutions. Therefore, in this example, the domain excludes \( x = -\frac{1}{2} \) and \( x = 1 \), keeping the expression well-defined.