Solve for x (x+6)/(x+3)=2-(5x+12)/(x+3)

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:

1. Prime factorize each number.

2. 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:

1. 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.

2. 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.

3. Distributive Property: A property of multiplication over addition or subtraction, stating that $a(b+c)=ab+ac$.

4. Simplifying Equations: The process of combining like terms and reducing equations to their simplest form to solve for the unknown variable.

5. 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.