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Chapter 7: Problem 10

For exercises 1-66, simplify. $$ \frac{3 x-12}{15} $$

Short Answer

Expert verified
The simplified form is \( \frac{x-4}{5} \).

Step by step solution

01

Factor the Numerator

First, factor out the greatest common factor (GCF) from the numerator. The GCF of the terms in the numerator, 3 and -12, is 3. Factoring this out, the numerator becomes: \(3(x - 4)\).
02

Simplify the Fraction

Now that the numerator is factored, the expression looks like this: \(\frac{3(x - 4)}{15}\). Next, simplify the fraction by dividing both the numerator and the denominator by their GCF, which is 3.
03

Divide by the GCF

Divide both the numerator and the denominator by 3: \(\frac{3(x - 4)}{15} = \frac{3(x - 4)}{3 * 5} = \frac{x - 4}{5}\).

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Key Concepts

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

Greatest Common Factor
The Greatest Common Factor (GCF) of two or more numbers is the largest number that divides each of them without leaving a remainder. In our example, we need to find the GCF of 3 and -12. The GCF is 3 because 3 is the largest number that divides both 3 and -12 evenly.
To factor out the GCF, we divide each term in the expression by the GCF and write the expression as a product of the GCF and another factor. For \(3x-12\), factoring out 3 gives us \(3(x-4)\).
Understanding the GCF is essential because it helps us simplify complex algebraic expressions, making them easier to work with and solve.
Factoring Expressions
Factoring involves writing an expression as a product of its factors. Factors are numbers or expressions that multiply together to produce a given number or expression.
For algebraic expressions, we often start by factoring out the GCF. In our example, the expression \(3x-12\) can be factored by taking out the GCF of 3, resulting in \(3(x-4)\).
Factoring is a crucial step in simplifying algebraic expressions because it allows us to break down a complex expression into simpler components. This is particularly useful in solving equations, simplifying fractions, and performing other algebraic operations.
Simplifying Fractions
Simplifying fractions means reducing them to their simplest form. This involves dividing the numerator and the denominator by their GCF.
For example, after factoring the numerator in \( \frac{3(x-4)}{15} \), we can simplify the fraction by dividing both the numerator and the denominator by their GCF, which is 3. This gives us \( \frac{3(x-4)}{3 \times 5} \). Dividing both the top and bottom by 3, we simplify the fraction to \ \frac{x-4}{5} \.
Simplifying fractions is an essential skill for making complex fractions more manageable and for solving many types of math problems more easily.

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Most popular questions from this chapter

For exercises 59-66, use the five steps. Assume that the rate of work does not change if done individually or together. The water from a garden hose turned on at full pressure fills a hot tub in \(45 \mathrm{~min}\). If the drain is open, the hot tub empties in \(62 \mathrm{~min}\). Find the amount of time to fill the hot tub with the drain open. Round to the nearest whole number.

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)

Medical researchers collected data on 272 patients who were hospitalized for at least 24 hours with the 2009 H1N1 influenza in the United States from April 2009 to mid-June 2009. One out of four of these patients were admitted to an intensive care unit. About 9 out of 20 patients were children under the age of 18 years. Find the number of patients who were children. Round to the nearest whole number. (Source: www.nejm.org, Nov. 12, 2009)

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.

The relationship of \(x\) and \(y\) is a direct variation. When \(x=1, y=6\). a. Find the constant of proportionality, \(k\). b. Write an equation that represents this direct variation. c. Find \(y\) when \(x=4\), d. Use slope-intercept graphing to graph this equation. e. Use the graph to find \(y\) when \(x=2\).
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