Problem 38
Question
Simplify the expression. $$ \frac{4}{x+4}-\frac{7}{5 x} $$
Step-by-Step Solution
Verified Answer
The simplified form of the given expression is \(\frac{13x - 28}{(x+4)5x}\)
1Step 1: Finding Common Denominator
The main requirement is finding a common denominator that both fractions can be expressed in terms of. The common denominator for the fractions \(\frac{4}{x+4}\) and \(\frac{7}{5x}\) is the product of \(x+4\) and \(5x\), i.e. \((x+4)5x\).
2Step 2: Express Fractions Using Common Denominator
Express the first fraction \(\frac{4}{x+4}\) with the common denominator \((x+4)5x\). Multiply its numerator and denominator by \(5x\), getting \( \frac{20x}{(x+4)5x} \). Do the same for the second fraction \(\frac{7}{5x}\), but multiply its numerator and denominator by (\(x+4\)), giving \( \frac{7(x+4)}{(x+4)5x} \).
3Step 3: Simplify the Expression
Subtract the second fraction \( \frac{7(x+4)}{(x+4)5x} \) from the first fraction \( \frac{20x}{(x+4)5x} \). This gives \( \frac{20x - 7(x+4)}{(x+4)5x} \). I.e., \( \frac{20x - 7x - 28}{(x+4)5x} \). This simplifies further to \(\frac{13x - 28}{(x+4)5x} \).
Key Concepts
Common DenominatorSubtracting FractionsSimplifying Fractions
Common Denominator
When working with algebraic expressions that involve fractions, one of the first steps is to make sure that all fractions involved have the same denominator, known as the common denominator. This is crucial because fractions can only be added, subtracted, or compared directly when they have the same denominator. To find a common denominator, we look for a multiple that is common to all denominators in the problem.
For example, if we have fractions with denominators of \(x+4\) and \(5x\), the common denominator would be their product, \(x+4)\cdot(5x)\), which ensures that each original denominator is a factor of this common denominator. An easy way to think about it is that you are creating a 'universal' measuring stick by which all the fractions can be measured, which then enables the other operations, such as addition or subtraction, to be carried out with ease.
For example, if we have fractions with denominators of \(x+4\) and \(5x\), the common denominator would be their product, \(x+4)\cdot(5x)\), which ensures that each original denominator is a factor of this common denominator. An easy way to think about it is that you are creating a 'universal' measuring stick by which all the fractions can be measured, which then enables the other operations, such as addition or subtraction, to be carried out with ease.
Subtracting Fractions
Subtracting fractions might seem tricky at first, but with a common denominator, it becomes much simpler. After you have converted your fractions to have the same denominator, the next step is to subtract the numerators while keeping the common denominator the same.
In the context of algebraic expressions, like in our example \( \frac{4}{x+4} - \frac{7}{5x} \), after finding the common denominator, we rewrite each fraction as equivalent fractions with this common denominator. We then subtract the numerators. So, the core idea is to understand that you're subtracting quantities of the same 'size' since the denominators are the same. Remember, we are only altering the top part—the numerators—when subtracting fractions, which signifies how much of that 'universal measuring stick' we have.
In the context of algebraic expressions, like in our example \( \frac{4}{x+4} - \frac{7}{5x} \), after finding the common denominator, we rewrite each fraction as equivalent fractions with this common denominator. We then subtract the numerators. So, the core idea is to understand that you're subtracting quantities of the same 'size' since the denominators are the same. Remember, we are only altering the top part—the numerators—when subtracting fractions, which signifies how much of that 'universal measuring stick' we have.
Simplifying Fractions
Simplifying fractions is the process of reducing them to their lowest terms. This entails dividing the numerator and the denominator by their greatest common factor so that the fractions becomes as simple as possible without changing its value.
In algebra, simplifying can also involve expanding and then combining like terms, as in our example: \( \frac{20x - 7(x+4)}{(x+4)5x} \), which simplifies to \( \frac{13x - 28}{(x+4)5x} \). This step is important because it can reveal further insights into the expression, such as factors that may cancel out, or it can make the expression easier to work with, whether for graphing, finding limits, or integrating in calculus. The goal is to make the expression as clear and concise as possible, without any unnecessary complexity.
In algebra, simplifying can also involve expanding and then combining like terms, as in our example: \( \frac{20x - 7(x+4)}{(x+4)5x} \), which simplifies to \( \frac{13x - 28}{(x+4)5x} \). This step is important because it can reveal further insights into the expression, such as factors that may cancel out, or it can make the expression easier to work with, whether for graphing, finding limits, or integrating in calculus. The goal is to make the expression as clear and concise as possible, without any unnecessary complexity.
Other exercises in this chapter
Problem 38
Simplify the expression. $$ \left(\frac{3 x^{2}}{56}\right)\left(\frac{3}{x}+\frac{5}{x}\right) $$
View solution Problem 38
Factor first, then solve the equation. Check your solutions. \(\frac{1}{y^{2}-16}-\frac{2}{y+4}=\frac{2}{y-4}\)
View solution Problem 38
Write the quotient in simplest form. $$\frac{3 x+12}{4 x} \div \frac{x+4}{2 x}$$
View solution Problem 38
Simplify the expression if possible. $$ \frac{12-5 x}{10 x^{2}-24 x} $$
View solution