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Chapter 7: Problem 8

For exercises \(5-48\), simplify. $$ \frac{14}{x-5}-\frac{9}{x-5} $$

Short Answer

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

Step by step solution

01

Identify the common denominator

Both fractions share the same denominator, which is . This common denominator is \(x-5\).
02

Combine the numerators

Since the denominators are the same, combine the numerators by subtracting: \( 14 - 9 \).
03

Perform the subtraction

Subtract 9 from 14 to get: \( 14 - 9 = 5 \).
04

Write the simplified fraction

Combine the result with the common denominator: \( \frac{5}{x-5} \)

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Key Concepts

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

common denominator
In order to subtract fractions, the denominators must be the same.
The denominator is the bottom number of the fraction.
In our example, both fractions have the same denominator: \( x-5 \).
This means you do not have to find a new common denominator.
Having the same denominator makes the fractions easier to work with.
Let's move on to combining the numerators.
combining numerators
Once we know the denominators are the same, we can focus on the numerators (the top numbers of the fractions).
Combining the numerators involves subtraction, which will also be explained in more detail soon.
For the example \( \frac{14}{x-5} - \frac{9}{x-5} \), combine the numerators by writing: \[ 14 - 9 \].
Essentially, you subtract the top numbers and keep the common denominator the same.
Now that we have done this, let's dive into the actual subtraction in our next section.
subtraction in fractions
Subtraction in fractions works just like regular subtraction. Once you have the same denominator, subtract the numerators: \[ 14 - 9 = 5 \].
You keep the common denominator \( x-5 \) unchanged.
This means the problem \( \frac{14}{x-5}-\frac{9}{x-5} \) simplifies to: \[ \frac{5}{x-5} \].
This is a crucial step in simplifying fractions.
But we are not completely done yet. The final step is to write the simplified fraction correctly.
simplified fraction
After performing the subtraction, it's time to write the simplified fraction.
Combining the result from the previous steps: the new numerator \( 5 \) and the common denominator \( x-5 \).
The simplified fraction is \[ \frac{5}{x-5} \].
Always double-check to ensure no further simplification is needed.
In this case, \( \frac{5}{x-5} \) is already fully simplified.
Remember, simplifying fractions makes them easier to understand and work with in later problems!

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Most popular questions from this chapter

For exercises \(65-68\), evaluate. $$ \sqrt{16} $$

If the force acting on an object is constant, the relationship of the mass of the object, \(x\), and the acceleration of the object, \(y\), is an inverse variation. When the mass is \(1000 \mathrm{~kg}\), the acceleration is \(\frac{4 \mathrm{~m}}{1 \mathrm{~s}^{2}}\). a. Find the constant of proportionality, \(k\). Include the units of measurement. b. Write an equation that represents this relationship. c. Find the acceleration when the mass is \(1500 \mathrm{~kg}\). Round to the nearest tenth.

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For exercises 11-30, (a) solve. (b) check. $$ \frac{13}{d}-\frac{5}{9}=\frac{1}{6} $$

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