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Chapter 8: Problem 215

$$ \frac{5 a}{a+3}-\frac{a+2}{a+6} $$

Short Answer

Expert verified
The simplified form is \frac{4a^2 + 25a - 6}{(a+3)(a+6)}\.

Step by step solution

01

Find a common denominator

To subtract the fractions, first find a common denominator. The denominators are \(a+3\) and \(a+6\). The least common denominator (LCD) is \( (a+3)(a+6) \).
02

Rewrite each fraction with the common denominator

Rewrite each fraction with the common denominator \( (a+3)(a+6) \): \[\frac{5a(a+6)}{(a+3)(a+6)} - \frac{(a+2)(a+3)}{(a+3)(a+6)} \]
03

Simplify each numerator

Expand and simplify the numerators: \[\frac{5a(a+6)}{(a+3)(a+6)} - \frac{(a+2)(a+3)}{(a+3)(a+6)} \Rightarrow \frac{5a^2 + 30a}{(a+3)(a+6)} - \frac{a^2 + 5a + 6}{(a+3)(a+6)} \]
04

Combine the fractions

Combine the fractions by subtracting the numerators: \[ \frac{5a^2 + 30a - (a^2 + 5a + 6)}{(a+3)(a+6)} \Rightarrow \frac{5a^2 + 30a - a^2 - 5a - 6}{(a+3)(a+6)} \]
05

Simplify the numerator

Simplify the numerator: \[ \frac{4a^2 + 25a - 6}{(a+3)(a+6)} \]

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Key Concepts

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

common denominator
When dealing with algebraic fractions, finding a common denominator is key. This allows you to add or subtract fractions easily. A common denominator is a shared multiple of the denominators. In our exercise, the denominators are \(a+3\) and \(a+6\). To find the common denominator, multiply the denominators together: \((a+3)(a+6)\). This shared base helps us combine the fractions correctly.
fraction subtraction
Subtracting fractions requires both fractions to have the same denominator. Once we have the common denominator, we rewrite each fraction. For example:
  • Rewrite \(\frac{5a}{a+3}\) as \(\frac{5a(a+6)}{(a+3)(a+6)}\).
  • Rewrite \(\frac{a+2}{a+6}\) as \(\frac{(a+2)(a+3)}{(a+3)(a+6)}\).
This sets both fractions to be over the common denominator \((a+3)(a+6)\). We can then proceed to subtract the numerators and keep the common denominator: \( \frac{5a(a+6)}{(a+3)(a+6)} - \frac{(a+2)(a+3)}{(a+3)(a+6)} \).
simplifying expressions
Simplifying algebraic expressions helps make our work cleaner and easier to understand. After rewriting the fractions, we expand and simplify the numerators. For instance, expanding \(5a(a+6)\) gives us \(5a^2 + 30a\). Similarly, expanding \((a+2)(a+3)\) leads to \(a^2 + 5a + 6\). Then, subtract these simplified numerators while keeping the common denominator: \(\frac{5a^2 + 30a - (a^2 + 5a + 6)}{(a+3)(a+6)}\). This yields: \(\frac{4a^2 + 25a - 6}{(a+3)(a+6)}\).
expanding polynomials
To get a clear and manageable expression, expand polynomials carefully. Expansion involves distributing each term in one polynomial by every term in another. For example:
  • Expanding \( (a+2)(a+3)\) gives us \( a \times a + a \times 3 + 2 \times a + 2 \times 3 \), which simplifies to \(a^2 + 5a + 6\).
  • Similarly, expanding \( 5a(a+6) \) results in \(5a^2 + 30a\).
Expanding simplifies subtraction or addition. Use expansion before combining terms, which keeps the expression easy to work with.

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Most popular questions from this chapter

Dana enjoys taking her dog for a walk, but sometimes her dog gets away and she has to run after him. Dana walked her dog for 7 miles but then had to run for 1 mile, spending a total time of 2.5 hours with her dog. Her running speed was 3 mph faster than her walking speed. Find her walking speed.

In the following exercises, solve uniform motion applications Jim's speedboat can travel 20 miles upstream against a \(3 \mathrm{mph}\) current in the same amount of time it travels 22 miles downstream with a 3 mph current speed. Find the speed of the Jim's boat.

In the following exercises, solve uniform motion applications Matthew jogged to his friend's house 12 miles away and then got a ride back home. It took him 2 hours longer to jog there than ride back. His jogging rate was 25 mph slower than the rate when he was riding. What was his jogging rate?

Write an inverse variation equation to solve the following problems. The time required to empty a tank varies inversely as the rate of pumping. It took Janet 5 hours to pump her flooded basement using a pump that was rated at 200 gpm (gallons per minute), (a) Write the equation that relates the number of hours to the pump rate. (b) How long would it take Janet to pump her basement if she used a pump rated at \(400 \mathrm{gpm} ?\)

Write an inverse variation equation to solve the following problems. Paul, a dentist, determined that the number of cavities that develops in his patient's mouth each year varies inversely to the \(\begin{array}{lll}\text { number of } & \text { minutes } & \text { spent }\end{array}\) brushing each night. His patient, Lori, had 4 cavities when brushing her teeth 30 seconds \((0.5\) minutes) each night. (a) Write the equation that relates the number of cavities to the time spent brushing. (b) How many cavities would Paul expect Lori to have if she had brushed her teeth for 2 minutes each night?
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