Solve the Rational Equation for x 1+5/(x+1)=-(5x)/(x+1)

Solution:

Step 1:

Remove $1$ from each side to get $\frac{5}{x + 1} = -\frac{5x}{x + 1} - 1$.

Step 2:

Identify the least common denominator (LCD) for the equation.

Step 2.1:

The LCD is equivalent to the least common multiple (LCM) of the denominators, which are $x + 1, x + 1,$ and $1$.

Step 2.2:

The LCM is the smallest number that each of the denominators can divide into without a remainder. To find it:

1. Prime factorize each number.

2. For each prime factor, take the highest power that appears in any of the numbers.

Step 2.3:

Since $1$ has only itself as a factor, it is not considered a prime number.

Step 2.4:

The LCM of $1, 1,$ and $1$ is simply $1$.

Step 2.5:

The factor for $x + 1$ is the expression itself, which occurs once.

Step 2.6:

The LCM of $x + 1$ and $x + 1$ is just $x + 1$.

Step 3:

Clear the fractions by multiplying every term in $\frac{5}{x + 1} = -\frac{5x}{x + 1} - 1$ by the LCD, which is $x + 1$.

Step 3.1:

Multiply $\frac{5}{x + 1}$ and $-\frac{5x}{x + 1} - 1$ by $x + 1$.

Step 3.2:

Simplify the left side by canceling out the common factors.

Step 3.2.1:

Remove the common $x + 1$ factor, resulting in $5 = -\frac{5x}{x + 1}(x + 1) - (x + 1)$.

Step 3.3:

Now, simplify the right side.

Step 3.3.1:

Simplify each term by canceling out the common $x + 1$ factor.

Step 3.3.1.1:

Move the negative sign to the numerator in $-\frac{5x}{x + 1}$.

Step 3.3.1.2:

Apply the distributive property to $-5x - (x + 1)$.

Step 3.3.1.3:

Combine like terms to get $5 = -5x - x - 1$.

Step 3.3.2:

Combine $-5x$ and $-x$ to get $5 = -6x - 1$.

Step 4:

Solve the simplified equation for $x$.

Step 4.1:

Rewrite the equation as $-6x - 1 = 5$.

Step 4.2:

Isolate terms with $x$ on one side by adding $1$ to both sides, resulting in $-6x = 6$.

Step 4.3:

Divide both sides by $-6$ to solve for $x$.

Step 4.3.1:

Divide $-6x$ and $6$ by $-6$.

Step 4.3.2:

Simplify both sides to find $x = -1$.

Step 5:

Check the solution by substituting $x = -1$ back into the original equation to ensure it does not result in a contradiction or undefined expression.

The final solution is $x = -1$, provided it satisfies the original equation.

Knowledge Notes:

1. Rational Equations: Equations that involve rational expressions, which are fractions containing polynomials in the numerator and denominator.

2. LCD (Least Common Denominator): The smallest common multiple of the denominators of two or more fractions. It is used to combine fractions into a single fraction.

3. LCM (Least Common Multiple): The smallest number that is a multiple of two or more numbers.

4. Prime Factorization: Breaking down a number into its prime number factors.

5. Distributive Property: A property that allows you to multiply a sum or difference by multiplying each addend separately and then add the products.

6. Solving Rational Equations: Often requires finding the LCD to eliminate fractions and then solving the resulting polynomial or linear equation.

7. Checking Solutions: It is important to substitute the found solutions back into the original equation to ensure they do not make any denominator zero, as this would make the solution invalid.