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Chapter 0: Problem 116

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$6+\frac{1}{x}=\frac{7}{x}$$

Short Answer

Expert verified
The given mathematical statement \(6 + \frac{1}{x} = \frac{7}{x}\) is false. The correct statement ought to be \(6 + \frac{1}{x} = \frac{6x+1}{x}\)

Step by step solution

01

First Step: Understanding the given equation

The given equation is \(6 + \frac{1}{x} = \frac{7}{x}\). The knee-jerk reaction might be to assume that we can just add the 6 inside the fraction, but unfortunately algebra doesn鈥檛 work that way. If we look at the right side of the equation, we see that it is an operation of division and not an addition, therefore it's not possible to directly add a whole number to a fraction.
02

Second Step: Checking the statement

Solving the given statement for x results in a falsehood, as dividing seven by any number does not equal six plus its reciprocal. Therefore, the original statement is false.
03

Third Step: Correcting the equation

We need to change the equation properly to make it a true statement. The correct form should be according to the properties of fractions, \(6+\frac{1}{x}=\frac{6x+1}{x}\), by multiplying 6 by x. This means the correct statement should be \(6+\frac{1}{x}=\frac{6x+1}{x}\).

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Most popular questions from this chapter

Will help you prepare for the material covered in the first section of the next chapter. If \(y=4-x^{2},\) find the value of \(y\) that corresponds to values of \(x\) for each integer starting with -3 and ending with 3

Place the correct symbol, \(>\) or \(<\), in the shaded area between the given numbers. Do not use a calculator. Then check your result with a calculator. \(a, 3^{\frac{1}{2}}\square 3^{1}\) b. \(\sqrt{7}+\sqrt{18} \square \sqrt{7+18}\)

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. A company manufactures and sells blank audiocassette tapes. The weekly fixed cost is \(\$ 10,000\) and it costs \(\$ 0.40\) to produce each tape. The selling price is \(\$ 2.00\) per tape. How many tapes must be produced and sold each week for the company to generate a profit?

Perform the indicated operations. Simplify the result, if possible. $$\frac{a b}{a^{2}+a b+b^{2}}+\left(\frac{a c-a d-b c+b d}{a c-a d+b c-b d} \div \frac{a^{3}-b^{3}}{a^{3}+b^{3}}\right)$$

In Exercises \(133-136\), determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$8^{-\frac{1}{4}}=-2$$
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