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Chapter 7: Problem 53

For exercises 43-58, (a) solve. (b) check. $$ \frac{2}{x}=0 $$

Short Answer

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Step by step solution

01

- Understand the Equation

The given equation is \( \frac{2}{x} = 0 \). This means we need to find the value of \( x \) that makes the fraction equal to zero.
02

- Recognize Property of Fractions

For a fraction \( \frac{a}{b} \), it equals zero if and only if the numerator \( a = 0 \) and the denominator \( b \) is not zero. Here, the numerator is 2, which is not zero.
03

- Conclusion

Since the numerator 2 is not zero, the equation \( \frac{2}{x} = 0 \) has no solution. A fraction with a non-zero numerator cannot equal zero, regardless of the denominator value.

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Key Concepts

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

Understanding Fractions
Fractions are a fundamental part of elementary algebra. In a fraction, such as \(\frac{a}{b}\), the top number is called the numerator and the bottom number is called the denominator. Fractions can represent parts of a whole or ratios between numbers.

To solve an equation involving fractions, it's essential to understand a key property: A fraction equals zero if the numerator is zero and the denominator is not zero. In our example, \(\frac{2}{x} = 0\), the numerator is 2. Since 2 is not zero, this fraction cannot be zero.

This leads us to conclude that no value of \x\ can make the fraction \(\frac{2}{x}\) equal to zero.
Solving Equations Involving Fractions
Solving equations is about finding the value of the unknown variable that makes the equation true. When fractions are involved, special rules apply. In the equation \(\frac{2}{x} = 0\), we need to find the value of \x\ such that the fraction equals zero.

For any fraction \(\frac{a}{b}\) to be zero, its numerator (a) must be zero because a non-zero number divided by any other number can never equal zero. Here, the numerator is 2, which is not zero. Therefore, the equation \(\frac{2}{x}\) will never equal zero, no matter what value \x\ takes.

This demonstrates how understanding the properties of fractions can make solving such equations straightforward.
Identifying No Solution in an Equation
Sometimes, you may encounter algebra problems that have no solution. This means there's no value you can substitute for the variable to make the equation true.

In our example, \(\frac{2}{x} = 0\), we found that there is no solution because a non-zero numerator cannot result in a fraction equaling zero. Recognizing a 'no solution' scenario involves identifying when the equation's conditions are impossible to satisfy.

By understanding the properties of fractions and how they interact in equations, you can determine when an equation has no solution. This saves time and avoids unnecessary computations.

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Most popular questions from this chapter

When the radiation is constant, the relationship of the current in an X-ray tube, \(x\), and the exposure time, \(y\), is an inverse variation. When the current is 600 milliamp, the exposure time is \(0.2 \mathrm{~s}\). Write an equation that represents this variation. Include the units.

For exercises 1-10, (a) solve. (b) check. $$ \frac{2}{3} x+\frac{3}{2}=\frac{1}{3} x+\frac{1}{6} $$

$$ \text { Solve: } 0.75=\frac{k}{60} $$

The relationship of \(x\) and \(y\) is an inverse variation. When \(x=2, y=6\). a. Find the constant of proportionality, \(k\). b. Write an equation that represents this inverse variation. c. Find \(y\) when \(x=4\).

For exercises \(67-82\), use the five steps and a proportion. In 2010 , there were \(14.9\) cases of syphilis per 100,000 Americans with a total of 45,834 cases of syphilis. Find the population of Americans used to create this ratio. Round to the nearest hundred. (Source: www.cdc.gov, 2011)
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