Problem 29
Question
Simplify each complex fraction. $$\frac{\frac{w+2}{w-1}-\frac{w-3}{w}}{\frac{w+4}{w}+\frac{w-2}{w-1}}$$
Step-by-Step Solution
Verified Answer
\( \frac{3(2w-1)}{2w^2+w-4} \)
1Step 1 - Simplify the Numerator
Simplify the expression in the numerator: \ \ \ \ \( \frac{w+2}{w-1} - \frac{w-3}{w} \) \ \ \ \ Find a common denominator for the terms: \ \ The common denominator is: \ \ \ \ \( (w-1)w \) \ \ Rewrite each term with the common denominator: \ \ \ \ \( \frac{(w+2)w}{(w-1)w} - \frac{(w-3)(w-1)}{w(w-1)} \) \ \ \ \ Combine the terms: \ \ \( \frac{w^2 +2w - (w^2 - 4w +3)}{(w-1)w} \) \ Simplify the expression: \ \ \( \frac{w^2 + 2w - w^2 + 4w - 3}{(w-1)w} \) \ \ \ \ \( \frac{6w -3}{(w-1)w} \) \ \ Factor out any common terms: \ \ \ \( \frac{3(2w -1)}{(w-1)w} \)
2Step 2 - Simplify the Denominator
Simplify the expression in the denominator: \ \ \ \( \frac{w+4}{w} + \frac{w-2}{w-1} \) \ \ Find a common denominator for the terms: The common denominator is: \ \ \( w(w-1) \) \ \ Rewrite each term with the common denominator: \ \ \ \( \frac{(w+4)(w-1)}{w(w-1)} + \frac{w(w-2)}{w(w-1)} \) \ \ Combine the terms: \ \ \( \frac{(w+4)(w-1) + w(w-2)}{w(w-1)} \) \ \ Expand and combine like terms: \ \ \ \( \frac{w^2 + 3w - 4 + w^2 - 2w}{w(w-1)} \) \ \( \frac{2w^2 + w -4}{w(w-1)} \)
3Step 3 - Simplify the Complex Fraction
Combine the simplified numerator and denominator: \ \ \( \frac{\frac{3(2w-1)}{w(w-1)}}{\frac{2w^2 + w -4}{w(w-1)}} \) \ Since the denominators are equal, they cancel out: \ \ \( \frac{3(2w-1)}{2w^2 + w -4} \)
Key Concepts
Numerator and Denominator SimplificationCommon DenominatorsFactoring ExpressionsCombining Like Terms
Numerator and Denominator Simplification
Begin by simplifying the numerator and the denominator separately. This helps in managing complex fractions.
Let's start with the **numerator**: \ \ \(\frac{w+2}{w-1} - \frac{w-3}{w}\)
We need a common denominator. In this case, the common denominator is \( (w-1)w \).
With the common denominator, rewrite each term as: \( \frac{(w+2)w}{(w-1)w} - \frac{(w-3)(w-1)}{w(w-1)} \)
Combine the terms: \( \frac{w^2 +2w - (w^2 - 4w +3)}{(w-1)w} \)
Simplify by removing the parenthesis and combining like terms: \( \frac{w^2 + 2w - w^2 + 4w - 3}{(w-1)w} \).
We get: \( \frac{6w -3}{(w-1)w} \).
Even in the denominator, we follow similar steps: \ \ \( \frac{w+4}{w} + \frac{w-2}{w-1} \)
The common denominator is \( w(w-1) \).
Rewrite each term to have the common denominator: \( \frac{(w+4)(w-1)}{w(w-1)} + \frac{w(w-2)}{w(w-1)} \)
Combine them: \( \frac{(w+4)(w-1) + w(w-2)}{w(w-1)} \)
Expand and simplify: \( \frac{w^2 + 3w - 4 + w^2 - 2w}{w(w-1)} \)
Resulting in: \( \frac{2w^2 + w -4}{w(w-1)} \).
Let's start with the **numerator**: \ \ \(\frac{w+2}{w-1} - \frac{w-3}{w}\)
We need a common denominator. In this case, the common denominator is \( (w-1)w \).
With the common denominator, rewrite each term as: \( \frac{(w+2)w}{(w-1)w} - \frac{(w-3)(w-1)}{w(w-1)} \)
Combine the terms: \( \frac{w^2 +2w - (w^2 - 4w +3)}{(w-1)w} \)
Simplify by removing the parenthesis and combining like terms: \( \frac{w^2 + 2w - w^2 + 4w - 3}{(w-1)w} \).
We get: \( \frac{6w -3}{(w-1)w} \).
Even in the denominator, we follow similar steps: \ \ \( \frac{w+4}{w} + \frac{w-2}{w-1} \)
The common denominator is \( w(w-1) \).
Rewrite each term to have the common denominator: \( \frac{(w+4)(w-1)}{w(w-1)} + \frac{w(w-2)}{w(w-1)} \)
Combine them: \( \frac{(w+4)(w-1) + w(w-2)}{w(w-1)} \)
Expand and simplify: \( \frac{w^2 + 3w - 4 + w^2 - 2w}{w(w-1)} \)
Resulting in: \( \frac{2w^2 + w -4}{w(w-1)} \).
Common Denominators
Finding a common denominator is crucial. It lets you combine fractions easily.
For the numerator, use the common denominator \( (w-1)w \).
Rewrite expressions as follows: \( \frac{(w+2)w}{(w-1)w} \) and \( \frac{(w-3)(w-1)}{w(w-1)} \). Combining gives: \( \frac{w^2 +2w - (w^2 - 4w +3)}{(w-1)w} \).
As you see, combining fractions with a common denominator simplifies calculations. For the denominator: \( \frac{(w+4)(w-1)}{w(w-1)} \) and \( \frac{w(w-2)}{w(w-1)} \) become one fraction. This method ensures both fractions have the same base, making it easier to combine terms.
Always look for a common factor to simplify fractions.
For the numerator, use the common denominator \( (w-1)w \).
Rewrite expressions as follows: \( \frac{(w+2)w}{(w-1)w} \) and \( \frac{(w-3)(w-1)}{w(w-1)} \). Combining gives: \( \frac{w^2 +2w - (w^2 - 4w +3)}{(w-1)w} \).
As you see, combining fractions with a common denominator simplifies calculations. For the denominator: \( \frac{(w+4)(w-1)}{w(w-1)} \) and \( \frac{w(w-2)}{w(w-1)} \) become one fraction. This method ensures both fractions have the same base, making it easier to combine terms.
Always look for a common factor to simplify fractions.
Factoring Expressions
Factoring helps simplify complex algebraic expressions.
For the numerator \( \frac{6w -3}{(w-1)w} \): Factor out the common term: \ 3(2w -1). So it becomes \( \frac{3(2w -1)}{(w-1)w} \).
This way, simplification becomes easier. For the denominator: \( \frac{2w^2 + w -4}{w(w-1)} \).
To factor: Look for patterns like quadratic structures. Factoring breaks down the expression into manageable parts.
Factoring reveals simpler forms that cancel out common terms.
For the numerator \( \frac{6w -3}{(w-1)w} \): Factor out the common term: \ 3(2w -1). So it becomes \( \frac{3(2w -1)}{(w-1)w} \).
This way, simplification becomes easier. For the denominator: \( \frac{2w^2 + w -4}{w(w-1)} \).
To factor: Look for patterns like quadratic structures. Factoring breaks down the expression into manageable parts.
Factoring reveals simpler forms that cancel out common terms.
Combining Like Terms
Combining like terms is essential to simplify fractions.
Look at the numerator: \( \frac{w^2 + 2w - w^2 + 4w - 3}{(w-1)w} \) \ \ Combine \ w \ terms: \ 6w - 3.
This leads to: \( \frac{6w -3}{(w-1)w} \).
Similarly, the denominator requires combining \ w \ terms: \( 2w^2 + w -4 \). Simplification follows these principles for both numerator and denominator.
By grouping and simplifying similar components, we achieve concise forms. This makes solving these expressions straightforward.
Look at the numerator: \( \frac{w^2 + 2w - w^2 + 4w - 3}{(w-1)w} \) \ \ Combine \ w \ terms: \ 6w - 3.
This leads to: \( \frac{6w -3}{(w-1)w} \).
Similarly, the denominator requires combining \ w \ terms: \( 2w^2 + w -4 \). Simplification follows these principles for both numerator and denominator.
By grouping and simplifying similar components, we achieve concise forms. This makes solving these expressions straightforward.
Other exercises in this chapter
Problem 29
Find the solution set to each equation. $$-\frac{5}{7}=\frac{2}{x}$$
View solution Problem 29
Solve each problem. Karen can ride her bike from home to school in the same amount of time as she can walk from home to the post office. She rides 10 miles per
View solution Problem 29
Reduce each rational expression to its lowest terms. $$\frac{-2 w^{2} x^{3} y}{6 w x^{5} y^{2}}$$
View solution Problem 30
Find the solution set to each equation. $$-\frac{3}{8}=\frac{5}{x}$$
View solution