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Chapter 0: Problem 16

Multiply or divide as indicated. $$\frac{6 x+9}{3 x-15} \cdot \frac{x-5}{4 x+6}$$

Short Answer

Expert verified
The product of the complex fractions after simplification is \(\frac{(3x+4.5)}{(2x+3)}\), excluding \(x=5\) and \(x=-1.5\) as possible solutions.

Step by step solution

01

Simplify Each Fraction

To start off, each fraction can be simplified by factoring out common terms. The first fraction \(\frac{6x+9}{3x-15}\) reduces to \(\frac{2(3x+4.5)}{3(x-5)}\). Taking out the common factor '3', it further simplifies to \(\frac{2}{1} \cdot \frac{(3x+4.5)}{(x-5)}\). Similarly, the second fraction \(\frac{x-5}{4x+6}\) simplifies to \(\frac{1}{2} \cdot \frac{(x-5)}{(2x+3)}\) after factoring out common terms.
02

Perform the Multiplication

Multiply the simplified fractions from Step 1. When multiplying fractions, simply multiply the numerators to get the new numerator, and multiply the denominators to get the new denominator. This will give the result \(\frac{2}{1} \cdot \frac{1}{2} \cdot \frac{(3x+4.5)(x-5)}{(x-5)(2x+3)}\). You can notice that \((x-5)\) occurs in the numerator and denominator, which can be cancelled out.
03

Simplify the Resulting Fraction

After cancelling out \((x-5)\), you're left with \(\frac{(3x+4.5)}{(2x+3)}\). Here it's important to remember that \(x\neq5\), and \(x\neq-1.5\), due to the fact they make the denominator zero. These values are considered extraneous and are excluded from the set of all possible values of x.

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Most popular questions from this chapter

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. Parts for an automobile repair cost \(\$ 175 .\) The mechanic charges \(\$ 34\) per hour. If you receive an estimate for at least \(\$ 226\) and at most \(\$ 294\) for fixing the car. what is the time interval that the mechanic will be working on the job?

a. A mathematics professor recently purchased a birthday cake for her son with the inscription $$ \text { Happy }\left(2^{\frac{5}{2}} \cdot 2^{\frac{3}{4}} \div 2^{\frac{1}{4}}\right) \text { th Birthday. } $$ How old is the son?b. The birthday boy, excited by the inscription on the cake, tried to wolf down the whole thing. Professor Mom, concerned about the possible metamorphosis of her son into a blimp, exclaimed, "Hold on! It is your birthday, so why not take \(\frac{8^{-\frac{4}{3}}+2^{-2}}{16^{-\frac{3}{4}}+2^{-1}}\) of the cake? I'll eat half of what's left over." How much of the cake did the professor eat?

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

Describe ways in which solving a linear inequality is different than solving a linear equation.

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.
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