Solve for x (x+6)/(x+3)=2-(5x+12)/(x+3)
The question presents you with an algebraic equation where a rational expression, (x+6)/(x+3), is set equal to another expression, 2-(5x+12)/(x+3). You are asked to find all the values of the variable x that satisfy this equation. To do that, you typically perform a series of algebraic manipulations to isolate x on one side of the equation in order to find its value(s).
$\frac{x + 6}{x + 3} = 2 - \frac{5 x + 12}{x + 3}$
Answer
Solution:
Step:1
Identify the least common denominator (LCD) for the fractions in the equation.
Step:1.1
To find the LCD, determine the least common multiple (LCM) of the denominators: $x + 3$, $1$, and $x + 3$.
Step:1.2
The LCM is the smallest number that each of the denominators divides into without a remainder. Follow these steps:
Prime factorize each number.
For each prime factor, take the highest power that appears in any of the numbers.
Step:1.3
Since $1$ has only itself as a factor, it is not considered a prime number.
Step:1.4
The LCM of $1$, $1$, and $1$ is simply $1$, as there are no prime factors to consider.
Step:1.5
The factor for $x + 3$ is the expression itself, occurring once: $(x + 3) = x + 3$.
Step:1.6
The LCM of $x + 3$ and $x + 3$ is their product, taken once: $x + 3$.
Step:2
Clear the fractions by multiplying every term in the equation $\frac{x + 6}{x + 3} = 2 - \frac{5x + 12}{x + 3}$ by the LCD, $x + 3$.
Step:2.1
Multiply each term by $x + 3$: $\frac{x + 6}{x + 3}(x + 3) = 2(x + 3) - \frac{5x + 12}{x + 3}(x + 3)$.
Step:2.2
Simplify the left-hand side of the equation.
Step:2.2.1
Eliminate the common factor of $x + 3$.
Step:2.2.1.1
Remove the common factor: $\frac{x + 6}{\cancel{x + 3}}(\cancel{x + 3}) = 2(x + 3) - \frac{5x + 12}{x + 3}(x + 3)$.
Step:2.2.1.2
Express the simplified equation: $x + 6 = 2(x + 3) - \frac{5x + 12}{x + 3}(x + 3)$.
Step:2.3
Simplify the right-hand side of the equation.
Step:2.3.1
Simplify each term individually.
Step:2.3.1.1
Apply the distributive property: $x + 6 = 2x + 2 \cdot 3 - \frac{5x + 12}{x + 3}(x + 3)$.
Step:2.3.1.2
Multiply $2$ by $3$: $x + 6 = 2x + 6 - \frac{5x + 12}{x + 3}(x + 3)$.
Step:2.3.1.3
Eliminate the common factor of $x + 3$.
Step:2.3.1.3.1
Move the negative sign to the numerator: $x + 6 = 2x + 6 + \frac{- (5x + 12)}{x + 3}(x + 3)$.
Step:2.3.1.3.2
Remove the common factor: $x + 6 = 2x + 6 + \frac{- (5x + 12)}{\cancel{x + 3}}(\cancel{x + 3})$.
Step:2.3.1.3.3
Rewrite the expression: $x + 6 = 2x + 6 - (5x + 12)$.
Step:2.3.1.4
Apply the distributive property: $x + 6 = 2x + 6 - 5x - 1 \cdot 12$.
Step:2.3.1.5
Multiply $5$ by $-1$: $x + 6 = 2x + 6 - 5x - 12$.
Step:2.3.1.6
Multiply $-1$ by $12$: $x + 6 = 2x + 6 - 5x - 12$.
Step:2.3.2
Combine like terms.
Step:2.3.2.1
Combine $2x$ and $-5x$: $x + 6 = -3x + 6 - 12$.
Step:2.3.2.2
Combine $6$ and $-12$: $x + 6 = -3x - 6$.
Step:3
Solve for $x$.
Step:3.1
Move all terms with $x$ to one side.
Step:3.1.1
Add $3x$ to both sides: $x + 6 + 3x = -6$.
Step:3.1.2
Combine $x$ and $3x$: $4x + 6 = -6$.
Step:3.2
Move constant terms to the other side.
Step:3.2.1
Subtract $6$ from both sides: $4x = -6 - 6$.
Step:3.2.2
Combine $-6$ and $-6$: $4x = -12$.
Step:3.3
Divide each side by $4$ to isolate $x$.
Step:3.3.1
Divide by $4$: $\frac{4x}{4} = \frac{-12}{4}$.
Step:3.3.2
Simplify the left side.
Step:3.3.2.1
Cancel the common factor of $4$: $\frac{\cancel{4}x}{\cancel{4}} = \frac{-12}{4}$.
Step:3.3.2.1.2
Divide $x$ by $1$: $x = \frac{-12}{4}$.
Step:3.3.3
Simplify the right side.
Step:3.3.3.1
Divide $-12$ by $4$: $x = -3$.
Step:4
Check that the solution does not make the original equation undefined. Since $x = -3$ does not result in a denominator of zero, it is a valid solution.
The final answer is $x = -3$.
Knowledge Notes:
Least Common Denominator (LCD): The LCD is the least common multiple of the denominators of a set of fractions. It is used to combine fractions into a single fraction or to eliminate fractions from an equation.
Least Common Multiple (LCM): The smallest number that is a multiple of two or more numbers. It is found by multiplying the highest powers of all prime factors present in the numbers.
Distributive Property: A property of multiplication over addition or subtraction, stating that $a(b + c) = ab + ac$.
Simplifying Equations: The process of combining like terms and reducing equations to their simplest form to solve for the unknown variable.
Checking Solutions: It is important to substitute the solution back into the original equation to ensure that it does not result in an undefined expression, such as division by zero.
