Simplify (x+1)/(x+2)+1/((x+1)/(x+2))

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:

1. 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.

2. Common Denominator: When adding or subtracting fractions, a common denominator is required. It is the product of the denominators of the fractions involved.

3. FOIL Method: A technique for multiplying two binomials. FOIL stands for First, Outer, Inner, Last, referring to the terms that are multiplied together.

4. 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.

5. Combining Like Terms: This involves adding or subtracting terms that have the same variable raised to the same power.

6. Simplifying Expressions: The process of reducing an expression to its simplest form by performing all possible operations and combining like terms.