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Chapter 7: Problem 15

Add or subtract. Write answer in lowest terms. \(\frac{5 m}{m+1}-\frac{1+4 m}{m+1}\)

Short Answer

Expert verified
\( \frac{m-1}{m+1} \)

Step by step solution

01

Identify common denominators

Both fractions have the same denominator \(m + 1\), so they can be directly subtracted.
02

Subtract the numerators

Subtract the numerator of the second fraction from the numerator of the first fraction: \( 5m - (1 + 4m)\).
03

Simplify the numerator

Distribute the negative sign and combine like terms: \( 5m - 1 - 4m = m - 1\).
04

Form the resulting fraction

Write the simplified numerator over the common denominator: \( \frac{m-1}{m+1} \).
05

Ensure the fraction is in lowest terms

Verify that \( \frac{m-1}{m+1} \) is already in its simplest form since the numerator and denominator have no common factors.

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Key Concepts

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

Common Denominators
When adding or subtracting fractions, it's crucial to have a common denominator. This is the same number or expression at the bottom part of your fractions. In this exercise, both fractions already have the same denominator, which is \(m + 1\). This makes subtraction easier because you can directly subtract the numerators.

To identify common denominators, follow these tips:
  • Look at the denominators of your fractions.
  • If they are the same, you can proceed with addition or subtraction.
If they are not the same, you will need to find a common denominator, usually by finding the least common multiple (LCM) of the denominators, before combining the fractions.
Simplify Fractions
Once you have subtracted or added the numerators over a common denominator, you need to simplify the fraction. Simplifying means expressing the fraction in its most basic form.

Here鈥檚 a quick guide on how to simplify fractions:
  • Combine like terms in the numerator.
  • Distribute negative signs, when necessary. In this case, subtract \(1 + 4m\) from \(5m\), giving you \(5m - 1 - 4m = m - 1\).
  • Write the simplified numerator over the common denominator.
This step ensures that your fraction is as simplified as possible before you go on to check for any further simplification or lowest terms.
Lowest Terms
The final step is to ensure that your fraction is in the lowest terms. A fraction is in its lowest terms when the numerator and the denominator have no common factors other than 1.

Here are the steps to check if your fraction is in the lowest terms:
  • Find the greatest common divisor (GCD) of the numerator and denominator.
  • If the GCD is 1, then the fraction is in its lowest terms.
  • In our example, \(\frac{m-1}{m+1} \) is already simplified since \(m-1\) and \(m+1\) have no common factors.
Always remember to check this step, as it ensures your solution is fully simplified and correct.

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Most popular questions from this chapter

Rewrite each rational expression with the indicated denominator. $$ \frac{2 p+3 q}{p^{2}+2 p q+q^{2}}=\frac{?}{(p+q)\left(p^{3}+q^{3}\right)} $$

Multiply or divide. Write each answer in lowest terms. See Examples \(3,6,\) and 7 . $$\frac{p^{2}+4 p-5}{p^{2}+7 p+10} \div \frac{p-1}{p+4}$$

Set up the equation you would use to solve each problem. Do not actually solve the equation. Working alone, Edward Good can paint a room in 8 hr. Abdalla Elusta can paint the same room working alone in 6 hr. How long will it take them if they work together? (Let \(t\) represent the time they work together.)

Solve each problem involving direct or inverse variation. If \(p\) varies inversely as \(q,\) and \(p=7\) when \(q=6,\) find \(p\) when \(q=2\)

If you are solving a formula for the letter \(k\), and your steps lead to the equation \(k r-m r=k m,\) what would be your next step?
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