Problem 33 - Algebra and Trigonometry

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. $$ \frac{2}{x}+3=\frac{5}{2 x}+\frac{13}{4} $$

Step-by-Step Solution

Answer

The solution to the given rational equation is \(x = -5/16 \) which holds true against the derived restriction 'x' cannot equal 0.

Step 1: Find Restrictions on Variable

The denominators in the given equation are 'x' and '2x'. For both cases, 'x' cannot be equal to zero (since we can't divide by zero). So 'x' ≠ 0 is the only restriction.

Step 2: Simplify the Equation

Next, clear the fractions by finding a common denominator which in this case is '2x'. Multiply every term in the equation by the common denominator to eliminate the fractions. This gives: \(2(2x) + 3(2x) = 5 + 13(2x).\ Refining further leads to: \(4x + 6x = 5 + 26x.\ Solving this equation, we have: \(10x = 5 + 26x.

Step 3: Solve for x

Rearrange the equation: \(10x - 26x = 5, which simplifies to: \(-16x = 5.\ Finally, solve for 'x' by dividing both sides by -16: \(x = -5/16.

Step 4: Re-check against Variable Restrictions

As the calculated value of 'x' is -5/16, which holds true against the derived restriction of 'x' ≠ 0, it is the valid solution of this equation.

Key Concepts

Variable Restrictions

When dealing with rational equations, it's essential to identify any variable restrictions. These restrictions occur because we cannot divide by zero. In this exercise, the denominators include 'x' and '2x'. To prevent division by zero, 'x' cannot equal zero. Therefore, our restriction on 'x' for this equation is simply:

• \( x eq 0 \)

This restriction ensures that all operations remain valid within the equation.

Common Denominator

Finding a common denominator is a key step when working with rational equations, as it allows you to eliminate fractions and simplify the solving process. For the given equation, the denominators 'x' and '2x' share a common factor. Consequently, the smallest common denominator we can use is '2x'. By multiplying every term in the equation by this common denominator:

• \( 2(2x) + 3(2x) = 5 + 13(2x) \)

we effectively remove the fractions, paving the way for simpler algebraic manipulation.

Solving Equations

Once you have a common denominator and the equation is simplified, you can start solving for the variable. In our case, multiplying each term by the common denominator '2x' yields:

• \( 4x + 6x = 5 + 26x \)

Reorganize the equation to bring like terms together. This leads to:

• \( 10x = 5 + 26x \)

Further simplification by getting all 'x' terms on one side gives:

• \( -16x = 5 \)

The value of 'x' can then be found by dividing through:

• \( x = -\frac{5}{16} \)

This solution reflects the steps of direct solving for 'x', prioritizing clarity and correctness.

Denominator Zero

One crucial aspect of rational equations is ensuring the denominator never equals zero, as division by zero is undefined. Originally, our variable restriction was determined to be \( x eq 0 \). With this restriction in mind, we solved the equation and found that \( x = -\frac{5}{16} \). It's imperative that this solution does not violate the established restriction.

• Since -\( \frac{5}{16} eq 0 \), it respects the restriction.

Therefore, \( x = -\frac{5}{16} \) is indeed a valid solution, ensuring both the mathematical soundness of the answer and the respect of the initial restrictions.