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Chapter 2: Problem 98

For exercises 97-100, simplify. $$ \frac{3}{4}(2 x+12) $$

Short Answer

Expert verified
\( \frac{3x}{2} + 9 \)

Step by step solution

01

Distribute the Fraction

Distribute the fraction \( \frac{3}{4} \) to each term inside the parentheses. This gives us \( \frac{3}{4} \) multiplied by \ 2x \ and \( \frac{3}{4} \) multiplied by 12.
02

Multiply the First Term

Multiply \( \frac{3}{4} \) by \ 2x \ to get \( \frac{3}{4} \times 2x = \frac{3 \times 2x}{4} = \frac{6x}{4} = \frac{3x}{2} \).
03

Multiply the Second Term

Multiply \( \frac{3}{4} \) by 12 to get \( \frac{3}{4} \times 12 = \frac{3 \times 12}{4} = \frac{36}{4} = 9 \).
04

Combine the Results

Combine the simplified terms from Steps 2 and 3 to get the final simplified expression: \( \frac{3x}{2} + 9 \).

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Key Concepts

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

Distributive Property
The distributive property is used for multiplying a single term across each term within a set of parentheses. It's like spreading the multiplication over each part.

For example, in the expression \(\frac{3}{4}(2x + 12)\), you need to distribute \(\frac{3}{4}\) to both \(2x\) and \(12\).

  • First, multiply \(\frac{3}{4}\) by \(2x\)
  • Then, multiply \(\frac{3}{4}\) by \(12\)
This helps break the problem into smaller, easier-to-handle pieces. By following this property correctly, you ensure that every term inside the parentheses is accounted for in the multiplication process.
Fraction Multiplication
Multiplying fractions might seem tricky, but it follows simple rules. To multiply fractions, multiply the numerators together and the denominators together.

In the context of our example, when multiplying \(\frac{3}{4}\) by \(2x\), we perform the following:

\[ \frac{3}{4} \times 2x = \frac{3 \times 2x}{4} \text{ or } \frac{6x}{4} \text{ or even more simplified to } \frac{3x}{2} \]
When multiplying \(\frac{3}{4}\) by 12, we perform the same steps:

\[ \frac{3}{4} \times 12 = \frac{3 \times 12}{4} \text{ or } \frac{36}{4} = 9 \]
Multiplying a fraction and an integer involves distributing the entire integer into the fraction's numerator. Then, simplify if possible.
Combining Like Terms
Once you've distributed and simplified your terms, it鈥檚 time to combine like terms, if any.

In our simplified expression \(\frac{3x}{2} + 9\), we observe that these terms are not
Simplifying Fractions
Simplifying fractions is all about making them as simple as possible. This means cancelling any common factors that appear in both the numerator and the denominator.

Take \(\frac{6x}{4}\) for example:
Both 6 and 4 are divisible by 2. By dividing each by 2, we get:

\[ \frac{6x}{4} = \frac{6/2 \times x}{4/2} = \frac{3x}{2} \]
This is the most simplified form of the fraction because no further common factors exist between 3 and 2.

It鈥檚 always good to double-check your work to ensure that your fractions are as simple as they can be.

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Most popular questions from this chapter

If the population of a city includes 75,000 people who are not immune to measles, find the minimum number of people who will get measles if they are exposed to the virus. Before measles immunization was available, nearly everyone in the U.S. got measles. ... Measles is one of the most infectious diseases in the world and is frequently imported into the U.S. ... More than 90 percent of people who are not immune will get measles if they are exposed to the virus. (Source: www.cdc.gov, Aug. 2010)

For exercises 93-96, the completed problem has one mistake. (a) Describe the mistake in words, or copy down the whole problem and highlight or circle the mistake. (b) Do the problem correctly. Problem: Solve: \(-2 x+12>36\) Incorrect Answer: \(-2 x+12>36\) $$ \begin{aligned} -12 &-12 \\ \hline-2 x+0 &>24 \\ \frac{-2 x}{-2} &>\frac{24}{-2} \\ x &>-12 \end{aligned} $$

Solve \(3 x+2 y=-8\) for \(y\).

For exercises 89-92, solve. Use a calculator to do the arithmetic. $$ \frac{c}{342}=-1840 $$

Solve \(V=\frac{L W T}{144}\) for \(W\).
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