Simplify (x+1)/(x+2)+1/((x+1)/(x+2))
The question is asking you to perform algebraic manipulation on a given complex rational expression. Specifically, you are to take the sum of a fraction, (x+1)/(x+2), and the reciprocal of that same fraction, 1/((x+1)/(x+2)), and simplify the result into a more straightforward or reduced form. Simplification may involve finding a common denominator, combining like terms, and reducing the expression wherever possible to express the final result in the simplest form.
$\frac{x + 1}{x + 2} + \frac{1}{\frac{x + 1}{x + 2}}$
Answer
Solution:
Step:1
Break down the expression into simpler components.
Step:1.1
Take the reciprocal of the second term and multiply it with the first term. $\frac{x + 1}{x + 2} + \frac{x + 2}{x + 1}$
Step:1.2
Rewrite $1$ as $\frac{x + 2}{x + 1}$ to combine the terms. $\frac{x + 1}{x + 2} + \frac{x + 2}{x + 1}$
Step:2
Create a common denominator for $\frac{x + 1}{x + 2}$ by multiplying it by $\frac{x + 1}{x + 1}$. $\frac{x + 1}{x + 2} \cdot \frac{x + 1}{x + 1} + \frac{x + 2}{x + 1}$
Step:3
Create a common denominator for $\frac{x + 2}{x + 1}$ by multiplying it by $\frac{x + 2}{x + 2}$. $\frac{x + 1}{x + 2} \cdot \frac{x + 1}{x + 1} + \frac{x + 2}{x + 1} \cdot \frac{x + 2}{x + 2}$
Step:4
Combine the terms over a common denominator of $(x + 2)(x + 1)$.
Step:4.1
Multiply $\frac{x + 1}{x + 2}$ by $\frac{x + 1}{x + 1}$. $\frac{(x + 1)(x + 1)}{(x + 2)(x + 1)} + \frac{x + 2}{x + 1} \cdot \frac{x + 2}{x + 2}$
Step:4.2
Multiply $\frac{x + 2}{x + 1}$ by $\frac{x + 2}{x + 2}$. $\frac{(x + 1)(x + 1)}{(x + 2)(x + 1)} + \frac{(x + 2)(x + 2)}{(x + 1)(x + 2)}$
Step:4.3
Rearrange the factors of $(x + 2)(x + 1)$. $\frac{(x + 1)(x + 1)}{(x + 1)(x + 2)} + \frac{(x + 2)(x + 2)}{(x + 1)(x + 2)}$
Step:5
Combine the numerators over the common denominator. $\frac{(x + 1)(x + 1) + (x + 2)(x + 2)}{(x + 1)(x + 2)}$
Step:6
Expand and simplify the numerator.
Step:6.1
Use the FOIL Method to expand $(x + 1)(x + 1)$.
Step:6.1.1
Distribute each term in the first binomial across the second. $\frac{x(x + 1) + 1(x + 1) + (x + 2)(x + 2)}{(x + 1)(x + 2)}$
Step:6.1.2
Distribute $x$ across $(x + 1)$. $\frac{x \cdot x + x \cdot 1 + 1(x + 1) + (x + 2)(x + 2)}{(x + 1)(x + 2)}$
Step:6.1.3
Distribute $1$ across $(x + 1)$. $\frac{x \cdot x + x \cdot 1 + 1 \cdot x + 1 \cdot 1 + (x + 2)(x + 2)}{(x + 1)(x + 2)}$
Step:6.2
Combine like terms and simplify.
Step:6.2.1
Simplify each term.
Step:6.2.1.1
Square $x$. $\frac{x^2 + x \cdot 1 + 1 \cdot x + 1 \cdot 1 + (x + 2)(x + 2)}{(x + 1)(x + 2)}$
Step:6.2.1.2
Multiply $x$ by $1$. $\frac{x^2 + x + 1 \cdot x + 1 \cdot 1 + (x + 2)(x + 2)}{(x + 1)(x + 2)}$
Step:6.2.1.3
Multiply $1$ by $1$. $\frac{x^2 + x + x + 1 + (x + 2)(x + 2)}{(x + 1)(x + 2)}$
Step:6.2.2
Add $x$ and $x$. $\frac{x^2 + 2x + 1 + (x + 2)(x + 2)}{(x + 1)(x + 2)}$
Step:6.3
Use the FOIL Method to expand $(x + 2)(x + 2)$.
Step:6.3.1
Distribute each term in the first binomial across the second. $\frac{x^2 + 2x + 1 + x(x + 2) + 2(x + 2)}{(x + 1)(x + 2)}$
Step:6.3.2
Distribute $x$ across $(x + 2)$. $\frac{x^2 + 2x + 1 + x \cdot x + x \cdot 2 + 2(x + 2)}{(x + 1)(x + 2)}$
Step:6.3.3
Distribute $2$ across $(x + 2)$. $\frac{x^2 + 2x + 1 + x \cdot x + x \cdot 2 + 2 \cdot x + 2 \cdot 2}{(x + 1)(x + 2)}$
Step:6.4
Combine like terms and simplify.
Step:6.4.1
Simplify each term.
Step:6.4.1.1
Square $x$. $\frac{x^2 + 2x + 1 + x^2 + x \cdot 2 + 2x + 2 \cdot 2}{(x + 1)(x + 2)}$
Step:6.4.1.2
Multiply $2$ by $x$. $\frac{x^2 + 2x + 1 + x^2 + 2 \cdot x + 2x + 2 \cdot 2}{(x + 1)(x + 2)}$
Step:6.4.1.3
Multiply $2$ by $2$. $\frac{x^2 + 2x + 1 + x^2 + 2x + 2x + 4}{(x + 1)(x + 2)}$
Step:6.4.2
Add $2x$ and $2x$. $\frac{x^2 + 2x + 1 + x^2 + 4x + 4}{(x + 1)(x + 2)}$
Step:6.5
Add $x^2$ and $x^2$. $\frac{2x^2 + 2x + 1 + 4x + 4}{(x + 1)(x + 2)}$
Step:6.6
Add $2x$ and $4x$. $\frac{2x^2 + 6x + 1 + 4}{(x + 1)(x + 2)}$
Step:6.7
Add $1$ and $4$. $\frac{2x^2 + 6x + 5}{(x + 1)(x + 2)}$
Knowledge Notes:
Reciprocal: The reciprocal of a number or a fraction is the inverse of it. For a fraction, you flip the numerator and denominator to find its reciprocal.
Common Denominator: When adding or subtracting fractions, a common denominator is required. It is the product of the denominators of the fractions involved.
FOIL Method: A technique for multiplying two binomials. FOIL stands for First, Outer, Inner, Last, referring to the terms that are multiplied together.
Distributive Property: This property states that $a(b + c) = ab + ac$. It is used to multiply a single term by each term inside a parenthesis.
Combining Like Terms: This involves adding or subtracting terms that have the same variable raised to the same power.
Simplifying Expressions: The process of reducing an expression to its simplest form by performing all possible operations and combining like terms.
