Solve the Rational Equation for x 1/(x+9)+3=-(4x)/(x+9)
Brief Explanation of the Question:
The problem is asking to find the value(s) of the variable x that will satisfy the given rational equation. A rational equation is one that contains at least one rational expression, which is a fraction whose numerator and/or denominator is a polynomial. The specific equation provided is a combination of two rational expressions and a constant: the first expression is 1/(x+9), the second term is the constant 3, and the third is -(4x)/(x+9). The task involves combining these expressions and solving for x while considering any restrictions on the variable that arise from the denominators of the rational expressions.
$\frac{1}{x + 9} + 3 = - \frac{4 x}{x + 9}$
Answer
Solution:
Step:1
Isolate the fraction on one side by subtracting $3$ from both sides of the given equation: $\frac{1}{x + 9} + 3 = -\frac{4x}{x + 9}$ becomes $\frac{1}{x + 9} = -\frac{4x}{x + 9} - 3$.
Step:2
Identify the Least Common Denominator (LCD) to combine the fractions.
Step:2.1
The LCD is equivalent to the Least Common Multiple (LCM) of the denominators: $x + 9$ and $1$.
Step:2.2
To find the LCM, list the prime factors and multiply each factor by the highest power it appears with among the numbers.
Step:2.3
Note that $1$ is not considered a prime number as it has only one divisor, itself.
Step:2.4
The LCM of $1$ is simply $1$ since it is the only factor.
Step:2.5
The term $x + 9$ is its own factor, and it appears once.
Step:2.6
The LCM of $x + 9$ and $x + 9$ is $x + 9$ since it is the only factor and it appears once.
Step:3
Clear the fractions by multiplying every term by the LCD, which is $x + 9$.
Step:3.1
Multiply the entire equation by $x + 9$: $\frac{1}{x + 9}(x + 9) = -\frac{4x}{x + 9}(x + 9) - 3(x + 9)$.
Step:3.2
Simplify the left side by canceling out the common factors.
Step:3.2.1
Eliminate the common $x + 9$ factor: $\frac{1}{\cancel{x + 9}}(\cancel{x + 9}) = -\frac{4x}{x + 9}(x + 9) - 3(x + 9)$.
Step:3.2.1.1
The equation simplifies to $1 = -\frac{4x}{x + 9}(x + 9) - 3(x + 9)$.
Step:3.3
Simplify the right side by distributing and combining like terms.
Step:3.3.1
Distribute and cancel out common factors.
Step:3.3.1.1
Move the negative sign to the numerator: $1 = -4x - 3(x + 9)$.
Step:3.3.1.2
Apply the distributive property: $1 = -4x - 3x - 27$.
Step:3.3.2
Combine like terms: $1 = -7x - 27$.
Step:4
Solve for $x$.
Step:4.1
Rewrite the equation: $-7x - 27 = 1$.
Step:4.2
Isolate the $x$ term by moving constants to the other side.
Step:4.2.1
Add $27$ to both sides: $-7x = 1 + 27$.
Step:4.2.2
Combine the constants: $-7x = 28$.
Step:4.3
Divide both sides by $-7$ to solve for $x$.
Step:4.3.1
Divide each term: $\frac{-7x}{-7} = \frac{28}{-7}$.
Step:4.3.2
Simplify both sides: $x = \frac{28}{-7}$.
Step:4.3.3
Calculate the division: $x = -4$.
Knowledge Notes:
Rational Equations: Equations that involve fractions where the numerator and/or the denominator are polynomials.
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.
LCM (Least Common Multiple): The smallest multiple that is exactly divisible by every number in a given set.
Prime Factorization: Breaking down a composite number into a product of its prime factors.
Distributive Property: A property that allows one to distribute a multiplied value across terms within parentheses, such as $a(b + c) = ab + ac$.
Combining Like Terms: The process of simplifying expressions by adding or subtracting terms that have the same variables raised to the same power.
Isolating the Variable: The process of manipulating an equation to get the variable of interest by itself on one side of the equation.
Solving Linear Equations: The process of finding the value of the variable that makes the equation true, which often involves operations such as adding, subtracting, multiplying, and dividing both sides of the equation by the same number.
