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Chapter 11: Problem 26

Solve the equation. $$\frac{1}{4}+\frac{4}{x}=\frac{1}{x}$$

Short Answer

Expert verified
The solution to the equation \(\frac{1}{4}+\frac{4}{x}=\frac{1}{x}\) is \(x = -12\).

Step by step solution

01

Eliminate the denominators

To make the equation easier to handle, first multiply every term by \(4x\), which is the least common multiplier. So the given equation \(\frac{1}{4}+\frac{4}{x}=\frac{1}{x}\) would be transformed to \(x+16=4\).
02

Solve for \(x\)

Rearrange the equation to find the solution for \(x\). It should look like this: \(x=4-16\).
03

Check the solution

The last step is to check if the solution \(x=-12\) is correct, by substituting \(x\) with \(-12\) in the original equation. You can confirm if \(\frac{1}{4}+\frac{4}{-12}=\frac{1}{-12}\). Checking will find that both sides of the equation are equal. So, \(x = -12\) is the correct solution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rational Equations
Rational equations involve fractions that have polynomials in the numerator, the denominator, or both. These equations can appear confusing at first, but they follow similar rules to solve as with any other algebraic equation.
The main challenge with rational equations is often dealing with the fractions. The presence of variables in the denominator requires careful handling.

To solve rational equations:
  • Identify any restrictions on the variable, usually stemming from denominators that cannot be zero.
  • Simplify the equation by finding a common denominator if necessary.
  • Clear the fractions by multiplying every term by the least common denominator鈥攖his transforms the equation into a polynomial form that's easier to solve.
Always remember, the solutions must not result in division by zero, so cross-check them against the original denominators at the end.
Common Denominators
The concept of common denominators is vital to solving rational equations. When you're working with fractions, it's often necessary to have the same denominator across all terms to simplify or to perform operations like addition or subtraction.

For rational equations, establishing a common denominator allows you to remove the fractions altogether. In our exercise, the least common denominator (LCD) is calculated by finding the smallest multiple that each of the individual denominators divides into.
In the exercise:
  • The denominators are 4 and x.
  • The LCD is obtained by combining these, resulting in 4x.
  • By multiplying the entire equation by the LCD, you effectively "clear" the fractions, leaving a simpler algebraic equation to solve.
This process not only simplifies the equation but also helps to focus on finding solutions without the distraction of fractional forms.
Checking Solutions
Once a solution for a rational equation is found, it鈥檚 crucial to verify its accuracy.Checking solutions ensures that the found answer is indeed correct and satisfies the original equation.

Here's how to verify the solution:
  • Substitute your solution for the variable back into the original equation.
  • Ensure both sides of the equation give the same numerical result.
  • Check that none of the denominators result in division by zero with the solutions found.
In our example, once we found that \(x = -12\), substituting this into the original equation showed equality on both sides.This confirmed that the solution is valid and doesn鈥檛 introduce any division by zero.
This final checking step is often overlooked but is crucial for correctly solving rational equations.

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Most popular questions from this chapter

Simplify the expression. (Review \(8.3 \text { for } 11.7)\) $$\frac{5}{10 x}$$

When you add rational expressions, you may need to factor a trinomial to find the LCD. Study the sample below. Then simplify the expressions in Exercises 46鈥�49. $$\text { Sample: } \frac{2 x}{x^{2}-1}+\frac{3}{x^{2}+x-2}=\frac{2 x}{(x+1)(x-1)}+\frac{3}{(x-1)(x+2)}$$ The LCD is \((x+1)(x-1)(x+2)\) Note: If you just used \(\left(x^{2}-1\right)\left(x^{2}+x-2\right)\) as the common denominator, the factor \((x-1)\) would be included twice. $$\frac{7 x+2}{16-x^{2}}+\frac{7}{x-4}$$

Simplify the expression. $$\frac{-8}{3 x^{2}}+\frac{11}{3 x^{2}}$$

Completely factor the expression. $$36 x^{5}-90 x^{3}$$

Write the equation in standard form. (Lesson 9.5 for 11.7 ) $$6 x^{2}=5 x-7$$
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