Problem 35 - College Algebra Essentials

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 = 2. The restriction is x ≠ 0.

Step 1: Identify Restrictions

To find restrictions on the variable x, we need to find values of x that make the denominator of a fraction equal to 0. Here in this equation, the denominators are 3x and 6x. Equating these to zero leads to x=0 for both. Therefore, the restriction on the variable x is that x is not equal to 0.

Step 2: Simplify the Equation

First, we eliminate the denominators by multiplying every term by the LCM of the denominators, which here is 6x. The equation becomes: 2*2 + 6x*1/4 = 11 - 6x*1/3. Simplifying further, we get 4 + 1.5x = 11 - 2x.

Step 3: Solve the Equation

To solve the equation, we move all terms including x to one side and the constants to the other: 1.5x + 2x = 11 - 4. This simplifies to 3.5x = 7. To find the value of x, we divide both sides by 3.5: x = 7/3.5 = 2. But remember that x can not be 0.

Key Concepts

Understanding Denominators

A rational equation often involves fractions, and fractions have denominators. The denominator is the bottom part of a fraction and indicates how many parts the whole is divided into. In a rational equation, like the one given where we have fractions like \(\frac{2}{3x}\) and \(\frac{11}{6x}\), the expression 3x and 6x are your denominators. They play a critical role in determining the behavior of the equation.

• Denominators tell us the "parts" based value of a fraction.
• Zero denominators make a fraction undefined, hence finding them is crucial.

So, understanding what makes these denominators zero is essential to solving rational equations safely and correctly.

Setting Restrictions

Finding restrictions is a key part of working with rational equations. A restriction comes into play when a denominator can be zero because division by zero is undefined.
In our equation \(\frac{2}{3x} + \frac{1}{4} = \frac{11}{6x} - \frac{1}{3}\), we identify potential points where denominators become zero.

• For \(3x = 0\) and \(6x = 0\), solving gives \(x = 0\).
• Therefore, \(x eq 0\) is a restriction.

Setting restrictions helps us avoid undefined values and ensures the solution is valid. Keeping these values in mind helps us solve equations without falling into math pitfalls.

Steps to Solving Equations

Solving rational equations involves a series of steps to simplify and eventually solve for the variable.

• First, determine any restrictions, as we did by finding when denominators become zero.
• Next, eliminate the denominators to simplify the equation. This is typically done by finding a common multiple.
• With denominators cleared, simplify the equation to combine like terms.
• Lastly, solve for the variable using basic algebraic manipulation.

For our given equation, once restricted values are noted, multiply each term by the Least Common Multiple (LCM) to eliminate fractions. Simplify to find a clean equation: \(4 + 1.5x = 11 - 2x\), then solve for \(x = 2\), matching restrictions.

The Importance of LCM (Least Common Multiple)

Finding the Least Common Multiple (LCM) is vital when solving rational equations since it allows us to clear fractions and simplifies the solving process.
Here's how the LCM works in this context:

• The LCM is the smallest number that is a multiple of two or more denominators.
• In our equation, the LCM of denominators \(3x\) and \(6x\) is \(6x\).
• Using \(6x\) as the common denominator eliminates fractions by balancing the equation.

By multiplying each term by the LCM, every term becomes easier to manage, allowing the equation to be simplified to a more solvable form. This efficient step is critical to getting results effectively and correctly.