Problem 16
Question
Perform the indicated operations. Reduce answers to their lowest terms. See Example \(I\) $$ \frac{x^{2}+3 x-3}{x-4}-\frac{x^{2}+4 x-7}{x-4} $$
Step-by-Step Solution
Verified Answer
-1
1Step 1 - Identify Common Denominator
Both fractions have the same denominator, which is \(x-4\).
2Step 2 - Subtract the Numerators
Write a single fraction by subtracting the numerators: \[ \frac{(x^2 + 3x - 3) - (x^2 + 4x - 7)}{x - 4} \]
3Step 3 - Simplify the Numerator
Distribute the subtraction and combine like terms: \[ \frac{x^2 + 3x - 3 - x^2 - 4x + 7}{x - 4} \]This simplifies to: \[ \frac{-x + 4}{x - 4} \]
4Step 4 - Simplify the Fraction
Factor out the negative sign in the numerator: \[ \frac{-(x - 4)}{x - 4} \]Cancel the \(x-4\) terms: \[ -1 \]
Key Concepts
subtraction of fractionssimplifying fractionsfactoring
subtraction of fractions
When subtracting fractions, the first step is to ensure they have a common denominator. This common base allows you to directly subtract the numerators. In our given problem, both fractions share the denominator \(x-4\), which simplifies the subtraction process.
To subtract these fractions, you need to subtract the numerators of the fractions while keeping the denominator the same. This operation can be represented as follows:
\[ \frac{(x^2 + 3x - 3) - (x^2 + 4x - 7)}{x - 4} \]
Make sure to distribute the subtraction properly and combine like terms. After distributing and simplifying, you'll get a single term in the numerator over the common denominator.
To subtract these fractions, you need to subtract the numerators of the fractions while keeping the denominator the same. This operation can be represented as follows:
\[ \frac{(x^2 + 3x - 3) - (x^2 + 4x - 7)}{x - 4} \]
Make sure to distribute the subtraction properly and combine like terms. After distributing and simplifying, you'll get a single term in the numerator over the common denominator.
simplifying fractions
Simplifying fractions involves reducing the expression to its simplest form. After we've subtracted the numerators in our problem, we get:
\[ \frac{x^2 + 3x - 3 - x^2 - 4x + 7}{x - 4} \],
When you combine like terms, this simplifies to:
\[ \frac{-x + 4}{x - 4} \], showing that terms can be further simplified by recognizing common factors or patterns.
Here, you can see that the numerator \( -x+4 \) and the denominator \( x-4 \) can be factored and eventually simplified.
\[ \frac{x^2 + 3x - 3 - x^2 - 4x + 7}{x - 4} \],
When you combine like terms, this simplifies to:
\[ \frac{-x + 4}{x - 4} \], showing that terms can be further simplified by recognizing common factors or patterns.
Here, you can see that the numerator \( -x+4 \) and the denominator \( x-4 \) can be factored and eventually simplified.
factoring
Factoring is a key step in simplifying algebraic fractions. In our example, after simplifying the numerator, we have:
\[ \frac{-x + 4}{x - 4} \].
We notice that \(-x+4\) can be rewritten by factoring out a negative sign to get:
\[ \frac{-(x - 4)}{x - 4} \].
Once factored, it becomes clear that the \(x-4\) terms can be canceled out. This leaves us with:
\[ -1 \]. Remember, factoring helps in identifying common terms that can be simplified, transforming the expression into a more manageable form.
\[ \frac{-x + 4}{x - 4} \].
We notice that \(-x+4\) can be rewritten by factoring out a negative sign to get:
\[ \frac{-(x - 4)}{x - 4} \].
Once factored, it becomes clear that the \(x-4\) terms can be canceled out. This leaves us with:
\[ -1 \]. Remember, factoring helps in identifying common terms that can be simplified, transforming the expression into a more manageable form.
Other exercises in this chapter
Problem 16
$$\text { Solve each formula for the indicated variable.}$$ $$\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}} \text { for } R_{2}$$
View solution Problem 16
Find the solution set to each equation. $$\frac{30}{x}=\frac{50}{x+10}+\frac{1}{2}$$
View solution Problem 16
Which real numbers cannot be used in place of the variable in each rational expression? $$\frac{3 b+1}{b^{2}-3 b-4}$$
View solution Problem 17
$$\text { Solve each formula for the indicated variable.}$$ $$\frac{P_{1} V_{1}}{T_{1}}=\frac{P_{2} V_{2}}{T_{2}} \text { for } T_{1}$$
View solution