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Chapter 7: Problem 65

For exercises 39-82, simplify. $$ \frac{3 p-1}{8 p} \div \frac{3 p^{2}+14 p-5}{6 p^{2}} $$

Short Answer

Expert verified
\(\frac{3 p}{4 (p + 5)}\)

Step by step solution

01

- Rewrite the division as multiplication by the reciprocal

Rewrite the division operation as multiplication by the reciprocal of the second fraction. This means turning \(\frac{3 p-1}{8 p} \div \frac{3 p^{2}+14 p-5}{6 p^{2}}\) into \(\frac{3 p-1}{8 p} \times \frac{6 p^{2}}{3 p^{2}+14 p-5}\).
02

- Factorize the quadratic expression

Factorize the quadratic expression in the denominator of the second fraction. \(3 p^{2} + 14 p - 5\) factors into \((3 p - 1)(p + 5)\).
03

- Substitute the factored expression

Replace the denominator \(3 p^{2} + 14 p - 5\) with \((3 p - 1)(p + 5)\). The expression becomes \(\frac{3 p-1}{8 p} \times \frac{6 p^{2}}{(3 p - 1)(p + 5)}\).
04

- Simplify the expression

Cancel the common factors in the numerator and denominator. The \(3 p - 1\) from the numerator of the first fraction cancels with \(3 p - 1\) in the denominator of the second fraction. Similarly, one \(p\) from \(6 p^{2}\) cancels with \(p\) in the denominator of the first fraction. This results in \(\frac{6 p}{8 (p + 5)}\).
05

- Simplify further

Simplify the fraction \(\frac{6 p}{8 (p + 5)}\) by dividing the numerator and denominator by their greatest common divisor, which is 2. This results in \(\frac{3 p}{4 (p + 5)}\).

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Key Concepts

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

Simplifying Fractions
To simplify a fraction, you need to reduce it to its simplest form. This means finding the greatest common divisor (GCD) of the numerator and the denominator, and then dividing both by that number. For instance, if you have \frac{6}{8}$$, the GCD of 6 and 8 is 2. Dividing both 6 and 8 by 2, we get \frac{3}{4}$$. Hence, \frac{6}{8}$$ simplifies to \frac{3}{4}$$.

Simplification also involves canceling out common factors in the numerator and denominator. For example, in the expression \frac{6p}{8}$$, we can cancel out a common factor of 2 to get \frac{3p}{4}$$.
Multiplying Rational Expressions
Multiplying rational expressions involves multiplying the numerators together and the denominators together. For example, when given two fractions \frac{a}{b}$$ and \frac{c}{d}$$, their product is \frac{a \times c}{b \times d}$$.

After multiplying, you should simplify the result by finding and canceling common factors between the numerator and denominator.

As illustrated in the exercise, \frac{3p-1}{8p} \times \frac{6p^{2}}{3p^{2}+14p-5}$$ changes to \frac{3p-1}{8p} \times \frac{6p^{2}}{(3p-1)(p+5)}$$. By canceling common factors, the expression became simpler.
Factoring Quadratic Expressions
Factoring quadratic expressions often simplifies solving and simplifying rational expressions. A quadratic expression typically has the form \textbf{ax^2 + bx + c}$$.

To factor it, look for two numbers that multiply to \textbf{ac}$$ and add to \textbf{b}$$. For example, to factor \textbf{3p^2 + 14p - 5}$$, we need numbers that multiply to -15 (3 * -5) and add to 14. These numbers are 15 and -1.

Thus, \textbf{3p^2 + 14p - 5}$$ factors into \textbf{(3p-1)(p+5)}$$, making it easier to handle in rational expressions.

This skill is crucial in simplifying and solving expressions, as seen in the exercise.

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Most popular questions from this chapter

For a fixed number of hotel rooms, the number of rooms cleaned per hour, \(x\), and the number of hours it takes to clean the rooms, \(y\), is an inverse variation. If a person can clean 8 rooms per hour, it takes 15 hr to clean the rooms. a. Find the constant of variation, \(k\). Include the units of measurement. b. Write an equation that represents this relationship. c. If a person can clean 6 rooms per hour, find the time needed to clean the rooms.

The relationship of the number of tickets sold, \(x\), and the total ticket receipts for an outdoor concert, \(y\), is a direct variation. When 11,000 tickets are sold, the total ticket receipts are \(\$ 495,000\). a. Find the constant of proportionality, \(k\). Include the units of measurement. b. Write an equation that represents this relationship. c. Find the number of tickets sold when the total ticket receipts are \(\$ 562,500\). d. Find the total ticket receipts from the sale of 7575 tickets. e. What does \(k\) represent in this equation?

For exercises 1-10, (a) solve. (b) check. $$ \frac{3}{5} x-\frac{1}{4}=\frac{9}{10} $$

For exercises \(67-82\), use the five steps and a proportion. Cyclosporine is an anti-rejection drug given to organ transplant patients. A bottle contains \(50 \mathrm{~mL}\) of liquid. Each milliliter of liquid contains \(100 \mathrm{mg}\) of cyclosporine. A kidney transplant patient needs to take \(850 \mathrm{mg}\) of cyclosporine each day. Find the amount of solution that the patient should take each day.

Find the number of \(2,200,000\) adults who binge drink about four times a month. Round to the nearest thousand. Binge drinking is a nationwide problem and bigger than previously thought. One in six adults binge drinks about four times a month. Binge drinking is defined as consuming four or more drinks for women or five or more drinks for men over a short period of time. Most binge drinkers are not alcohol-dependent. (Source: www.cdc.gov, Jan. 2012)
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