Solve for x 3/(6x-48)+1=3/(x-8)

Solution:

Step 1:

Isolate the fraction on one side by subtracting 1 from both sides: $\frac{3}{6x - 48} = \frac{3}{x - 8} - 1$.

Step 2:

Commence factoring.

Step 2.1:

Extract the common factor of 6 from $6x - 48$.

Step 2.1.1:

Factor out 6 from $6x$: $\frac{3}{6(x) - 48} = \frac{3}{x - 8} - 1$.

Step 2.1.2:

Factor out 6 from $-48$: $\frac{3}{6x - 6(8)} = \frac{3}{x - 8} - 1$.

Step 2.1.3:

Combine the factored terms: $\frac{3}{6(x - 8)} = \frac{3}{x - 8} - 1$.

Step 2.2:

Simplify the fraction by canceling out common factors.

Step 2.2.1:

Factor out 3 from the numerator: $\frac{3(1)}{6(x - 8)} = \frac{3}{x - 8} - 1$.

Step 2.2.2:

Factor out 3 from the denominator: $\frac{3(1)}{3(2(x - 8))} = \frac{3}{x - 8} - 1$.

Step 2.2.3:

Cancel out the common factor of 3: $\frac{\cancel{3} \cdot 1}{\cancel{3}(2(x - 8))} = \frac{3}{x - 8} - 1$.

Step 2.2.4:

Rewrite the simplified expression: $\frac{1}{2(x - 8)} = \frac{3}{x - 8} - 1$.

Step 3:

Determine the least common denominator (LCD) of the terms.

Step 3.1:

The LCD is found by identifying the least common multiple (LCM) of the denominators: $2(x - 8), x - 8, 1$.

Step 3.2:

The LCM is the smallest number that all terms divide into evenly. This involves listing prime factors and multiplying each factor the greatest number of times it occurs in any number.

Step 3.3:

Recognize that 2 is a prime number.

Step 3.4:

Acknowledge that 1 is not prime as it has only one positive factor, itself.

Step 3.5:

The LCM of 2, 1, and 1 is 2, as it's the product of all prime factors at their highest occurrence.

Step 3.6:

The factor for $x - 8$ is itself, occurring once.

Step 3.7:

The LCM of $x - 8$ and $x - 8$ is $x - 8$.

Step 3.8:

The LCD, being the smallest multiple of the numbers, is $2(x - 8)$.

Step 4:

Eliminate fractions by multiplying each term by the LCD, $2(x - 8)$.

Step 4.1:

Apply the LCD to each term: $\frac{1}{2(x - 8)} \cdot 2(x - 8) = \frac{3}{x - 8} \cdot 2(x - 8) - 2(x - 8)$.

Step 4.2:

Simplify the left side by canceling out common factors.

Step 4.2.1:

Use the commutative property: $2 \cdot \frac{1}{2(x - 8)}(x - 8) = \frac{3}{x - 8} \cdot 2(x - 8) - 2(x - 8)$.

Step 4.2.2:

Cancel the common factor of 2: $\cancel{2} \cdot \frac{1}{\cancel{2}(x - 8)}(x - 8) = \frac{3}{x - 8} \cdot 2(x - 8) - 2(x - 8)$.

Step 4.2.3:

Cancel the common factor of $x - 8$: $\frac{1}{\cancel{x - 8}}(\cancel{x - 8}) = \frac{3}{x - 8} \cdot 2(x - 8) - 2(x - 8)$.

Step 4.3:

Simplify the right side by distributing and combining like terms.

Step 4.3.1:

Apply the distributive property: $1 = 6 - 2(x - 8)$.

Step 4.3.2:

Combine like terms: $1 = 6 - 2x + 16$.

Step 5:

Solve for $x$ by isolating the variable.

Step 5.1:

Rearrange the equation: $-2x + 22 = 1$.

Step 5.2:

Move constant terms to the opposite side: $-2x = 1 - 22$.

Step 5.3:

Divide by the coefficient of $x$ to solve: $x = \frac{-21}{-2}$.

Step 6:

Express the solution in various forms.

Exact Form: $x = \frac{21}{2}$

Decimal Form: $x = 10.5$

Mixed Number Form: $x = 10 \frac{1}{2}$

Knowledge Notes:

The problem-solving process involves several key mathematical concepts:

1. Algebraic Manipulation: The ability to rearrange and simplify equations is fundamental to solving algebraic problems.

2. Factoring: This is the process of breaking down an expression into its constituent factors. It is often used to simplify fractions and solve equations.

3. Least Common Multiple (LCM) and Least Common Denominator (LCD): The LCM of a set of numbers is the smallest number that is a multiple of all the numbers in the set. The LCD is the LCM of the denominators of a set of fractions, used to combine fractions or eliminate them from an equation.

4. Distributive Property: This property allows for the multiplication of a number by a sum or difference, distributing the multiplication over all terms within the parentheses.

5. Cancelling Common Factors: When a factor appears in both the numerator and denominator of a fraction, it can be cancelled out, simplifying the fraction.

6. Commutative Property of Multiplication: This property states that the order in which two numbers are multiplied does not affect the product.

7. Prime Numbers: A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. Recognizing primes is essential in factoring.

8. Solving Linear Equations: The process of finding the value of the variable that makes the equation true. This often involves isolating the variable on one side of the equation.

By understanding and applying these concepts, one can solve a wide range of algebraic problems.