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Chapter 0: Problem 113

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\frac{x^{2}-25}{x-5}=x-5$$

Short Answer

Expert verified
The given statement is indeed false. To correct it, the equation should be \(\frac{x^{2}-25}{x-5}=x+5\).

Step by step solution

01

Simplify the left side

The given equation is \(\frac{x^{2}-25}{x-5}=x-5\). The numerator of the fraction on the left side can be factored using the difference of squares formula, which states that \(a^{2}-b^{2}=(a+b)(a-b)\). Using this, our equation becomes \(\frac{(x+5)(x-5)}{x-5}=x-5.
02

Simplify further

Now, cancel out the \(x-5\) terms that appear in both the numerator and the denominator of the left side of the equation. This leaves the equation as \(x+5=x-5\).
03

Analyze the result

Observing the equation \(x+5=x-5\), we can see a contradiction has arisen. This is not a valid equation because there is no real value for x that would satisfy this equation. Hence, the given equation, we started with, is false.
04

Correct the false equation

To make the given equation true, it should be \(\frac{x^{2}-25}{x-5}=x+5\). After simplifying, both sides of the equation will be equal and therefore the statement becomes true.

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Most popular questions from this chapter

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Because I want to solve \(25 x^{2}-169=0\) fairly quickly, I'll use the quadratic formula.

Explain how to simplify a rational expression.

Explain how to add rational expressions having no common factors in their denominators. Use \(\frac{3}{x+5}+\frac{7}{x+2}\) in your explanation.

Describe ways in which solving a linear inequality is similar to solving a linear equation.

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?
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