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Chapter 7: Problem 78

For exercises 39-82, simplify. $$ \frac{b^{2}+7 b-18}{b-2} \div \frac{b+9}{b+4} $$

Short Answer

Expert verified
\(\frac{b+4}{b-2}\)

Step by step solution

01

Factor the numerator of the first fraction

First, factor the quadratic expression in the numerator of the first fraction: \(\frac{b^{2}+7b-18}{b-2}\). To factor \(b^{2}+7b-18\), find two numbers that multiply to -18 and add to 7. These numbers are 9 and -2, so the factored form is \((b+9)(b-2)\):\[\frac{(b+9)(b-2)}{b-2}\].
02

Simplify the first fraction

Simplify the first fraction by canceling out the common term \((b-2)\). This leaves:\[\frac{(b+9)(b-2)}{b-2} = b+9\].
03

Set up the division of the fractions

Rewrite the division of the fractions as multiplication by the reciprocal. The original problem is:\[\frac{b+9}{b-2} \times \frac{b+4}{b+9}\].
04

Cancel out common factors

Notice that \(b+9\) appears in both the numerator of the first fraction and the denominator of the second fraction. Cancel out \(b+9\) to get:\[b+9 \times \frac{b+4}{b+9} = \frac{b+4}{b-2}\].

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Key Concepts

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

Factoring Quadratics
When simplifying algebraic expressions, factoring quadratics is often an essential step.
For example, consider the quadratic expression in the numerator \(b^{2} + 7b - 18\).
To factor this, we look for two numbers that multiply to -18 and add to 7. Those numbers are 9 and -2. Thus, the factored form of \(b^{2} + 7b - 18\) is \((b+9)(b-2)\).
Factoring quadratics breaks down complex polynomials into simpler multiplicative components, making them easier to handle in equations.
Canceling Common Factors
After factoring, the next step often involves simplifying the expression by canceling common factors.
In the given problem, we have the factored form \frac{(b+9)(b-2)}{b-2}\. Notice that the term \(b-2\) appears in both the numerator and the denominator.
By canceling out the common term \(b-2\), we simplify the fraction to \(b+9\).
This process reduces the expression to its simplest form, making it much easier to work with.
Division of Fractions
In algebra, division of fractions is simplified by multiplying by the reciprocal.
Given \frac{b+9}{b-2} \div \frac{b+9}{b+4}\, we rewrite this as \frac{b+9}{b-2} \times \frac{b+4}{b+9}\.
This converts the division into multiplication, making it straightforward to handle.
In our example, we notice that \(b+9\) is a common factor in the numerator of the first fraction and the denominator of the second fraction.
Cancelling \(b+9\) from both, we are left with the simplified expression \frac{b+4}{b-2}\.
Understanding this concept helps in performing operations on fractions efficiently.

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Most popular questions from this chapter

For exercises \(67-82\), use the five steps and a proportion. Find the number of 725,000 women in their mid \(-40 \mathrm{~s}\) with a history of normal pregnancy who would be expected to have a heart attack or stroke some 10 years later. Of 100 women in their mid-40's with a history of normal pregnancy, about 4 would be expected to have a heart attack or stroke some 10 years later. (Source: www.nytimes.com, March 17, 2009)

For a fixed length of household copper wire, the relationship of the cross- sectional area, \(x\), and the resistance, \(y\), is an inverse variation. When the cross-sectional area is \(3.14 \times 10^{-6} \mathrm{~m}^{2}\), the resistance is \(5.4 \times 10^{-3} \mathrm{ohm}\). a. Find the constant of proportionality, \(k\). Use scientific notation. Include the units of measurement. b. Write an equation that represents this relationship. c. Find the resistance when the cross-sectional area is \(2.05 \times 10^{-6} \mathrm{~m}^{2}\). Round the mantissa to the nearest tenth.

If the annual credit sales are constant, the relationship of the accounts receivable, \(x\), and the financial ratio receivables turnover, \(y\), is an inverse variation. The accounts receivable of a company are \(\$ 150\) million, and its receivables turnover ratio is 12 . a. Find the constant of proportionality. Include the units of measurement. b. Write an equation that represents this relationship. c. Find the receivables turnover ratio when the accounts receivable are \(\$ 200\) million.

The formula \(R=\frac{V}{I}\) represents the relationship of the resistance \(R\), voltage \(V\), and current \(I\) in an electric circuit. Assume that \(V\) is constant. Is the relationship of \(R\) and \(I\) a direct variation or an inverse variation?

The relationship of the time a tour guide works, \(x\), and the cost to hire the tour guide, \(y\), is a direct variation. When a tour guide works for \(15 \mathrm{hr}\), the cost is \(\$ 1125\). a. Find the constant of proportionality, \(k\). Include the units of measurement. b. Write an equation that represents this relationship. c. Find the cost to hire a tour guide for \(8 \mathrm{hr}\). d. What does \(k\) represent in this equation?
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