Problem 35 - College Algebra

Question

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. $$ \frac{2}{3 x}+\frac{1}{4}-\frac{11}{6 x}-\frac{1}{3} $$

Step-by-Step Solution

Answer

The solution to the equation is \(x=-18\), with the restriction that \(x\) cannot be \(0\).

Step 1: Identify Restrictions on the Variable

Identify the values of the variable that make the denominator zero. The denominators here are \(3x\) and \(6x\). The restriction on the variable occurs when \(x = 0\) because we cannot divide by zero.

Step 2: Simplify the equation

The given equation is \[\frac{2}{3x}+\frac{1}{4}-\frac{11}{6x}-\frac{1}{3}.\]The first and third terms of this equation have the same denominator after simplification, \(3x\). So, we can combine these two terms. Adding the numerators, we get \(\frac{2-22}{3x}\) or \(-\frac{20}{3x}\) simplify similarly the second and fourth term we get \(\frac{1}{4}-\frac{1}{3}=\frac{1*3 - 1*4}{4*3}\), that gives us \(-\frac{1}{12}\). Now the equation is simplified to \[-\frac{20}{3x}-\frac{1}{12}.\]

Step 3: Solve the equation

We are now at a stage where the equation is simplified. However, this equation equals 0. Therefore, \(-\frac{20}{3x}= \frac{1}{12}\) or, after cross multiplication and simplifying, we get \(x=-18\). But keep in mind that \(x\) does not equal \(0\), given the initial denominator.

Key Concepts

Variables in Denominators

Rational equations often contain variables in the denominator, making things a bit trickier to solve. This is due to the fact that division by zero is undefined. For example, in the original exercise, we have rational expressions like \( \frac{2}{3x} \) and \( \frac{11}{6x} \). These expressions indicate that if \( x = 0 \), the denominators would become zero, resulting in undefined expressions. Thus, it's important to identify these variables in the denominator, as they determine the restrictions that will guide how we solve the equation. Identifying these variables also helps ensure we avoid any mathematical mishaps later on. Let's make this concept clear:

• Look at each denominator carefully.
• Determine the value of the variable that would turn any denominator into zero.
• List these values as they become restrictions on your solution.

Understanding these restrictions is crucial for successfully navigating rational equations.

Solving Equations

Once we have the limitations from our denominators, the next step is solving the rational equation itself. Think of solving it as putting together a puzzle. Start by simplifying the fractions where possible. In the original exercise, the fractions need to be consolidated by finding a common denominator. However, it becomes straightforward when like fractions are combined. So, we simplify

• The terms \( \frac{2}{3x} \) and \( \frac{11}{6x} \) initially.
• Consolidate these by finding a single common denominator.
• Repeat this process for any other terms like \( \frac{1}{4} \) and \( \frac{1}{3} \).

Once simplified, the equation usually reduces to a more manageable form – allowing you to better isolate the variable and solve the equation. Remember, cross multiplication can be a helpful tool when dealing with equations involving fractions.

Restrictions on Variables

Determining restrictions is a crucial part of solving rational equations because they guide whether a solution is acceptable or not. As previously mentioned, these restrictions occur when a denominator could potentially become zero. In our example, since the denominators are \( 3x \) and \( 6x \), they become zero when \( x = 0 \). It means any solution that results in \( x = 0 \) must be discarded because it would invalidate the equation by introducing undefined elements.

• Always identify potential restrictions before solving.
• Check against these restrictions after finding a solution to see if it's valid.
• Remember, even though the process simplifies the equation, always revisit your restrictions to ensure the solution stays within those boundaries.

By keeping these restrictions in mind, you're not just solving the equation but also ensuring that your solution is mathematically sound.

Algebraic Simplification

The power of algebraic simplification cannot be understated when solving rational equations. This process involves combining like terms, clearing fractions, and reducing the equation to its simplest form. In our given exercise, it meant breaking down the given fractions to simplify the equation. We combined like terms and worked towards reducing the equation from complex fractions to something simpler to solve.

• Simplify by combining like terms wherever possible.
• Use cross multiplication to eliminate fractions and simplify calculations.
• Ensure each step maintains the balance of the equation.

Doing so not only helps reveal the solution more clearly but also makes checking your answer against the initial restrictions easier. Algebraic simplification is fundamental to making rational equations approachable and solvable.