Solve for x 9/(6x-9)=x/(2x-3)-1/3

Solution:

Step 1: Simplify each fraction by factoring.

Step 1.1: Extract the common factor from the denominator $6x - 9$.

Step 1.1.1: Factor out $3$ from $6x$ to get $\frac{9}{3(2x) - 9} = \frac{x}{2x - 3} - \frac{1}{3}$.

Step 1.1.2: Factor out $3$ from $-9$ to obtain $\frac{9}{3(2x) + 3(-3)} = \frac{x}{2x - 3} - \frac{1}{3}$.

Step 1.1.3: Combine the factored terms to simplify $\frac{9}{3(2x - 3)} = \frac{x}{2x - 3} - \frac{1}{3}$.

Step 1.2: Cancel out the common factor in the fraction $\frac{9}{3(2x - 3)}$.

Step 1.2.1: Factor $3$ from the numerator $9$ to get $\frac{3 \cdot 3}{3(2x - 3)} = \frac{x}{2x - 3} - \frac{1}{3}$.

Step 1.2.2: Eliminate the common factor to simplify $\frac{\cancel{3} \cdot 3}{\cancel{3}(2x - 3)} = \frac{x}{2x - 3} - \frac{1}{3}$.

Step 1.2.3: Rewrite the simplified expression as $\frac{3}{2x - 3} = \frac{x}{2x - 3} - \frac{1}{3}$.

Step 2: Identify the least common denominator (LCD) for the equation.

Step 2.1: The LCD is found by determining the least common multiple (LCM) of the denominators $2x - 3$, $2x - 3$, and $3$.

Step 2.2: To find the LCM, list the prime factors and multiply each factor by the greatest number of times it appears in any of the numbers.

Step 2.3: Recognize that $1$ is not prime because it has only one positive divisor, itself.

Step 2.4: Since $3$ is only divisible by $1$ and itself, it is a prime number.

Step 2.5: The LCM of $1, 1, 3$ is the product of all prime factors at their highest occurrence, which is $3$.

Step 2.6: The factor for $2x - 3$ is itself, occurring once.

Step 2.7: The LCM of $2x - 3, 2x - 3$ is simply $2x - 3$.

Step 2.8: The LCD, being the smallest multiple common to the numbers, is $3(2x - 3)$.

Step 3: Clear the fractions by multiplying each term by the LCD $3(2x - 3)$.

Step 3.1: Apply the LCD to each term in the equation $\frac{3}{2x - 3} = \frac{x}{2x - 3} - \frac{1}{3}$.

Step 3.2: Simplify the left side of the equation.

Step 3.2.1: Use the commutative property to rearrange the terms.

Step 3.2.2: Multiply $3$ by $\frac{3}{2x - 3}$.

Step 3.2.3: Cancel the common denominator $2x - 3$.

Step 3.2.3.1: Remove the common factor to get $9 = \frac{x}{2x - 3}(3(2x - 3)) - \frac{1}{3}(3(2x - 3))$.

Step 3.3: Simplify the right side of the equation.

Step 3.3.1: Simplify each term individually.

Step 3.3.1.1: Use the commutative property to rearrange the terms.

Step 3.3.1.2: Combine $3$ with $\frac{x}{2x - 3}$.

Step 3.3.1.3: Cancel the common denominator $2x - 3$.

Step 3.3.1.4: Simplify the term involving the factor of $3$.

Step 3.3.1.5: Apply the distributive property to expand the expression.

Step 3.3.1.6: Perform the subtraction to combine like terms.

Step 3.3.2: Combine the terms to get a simplified expression.

Step 4: Solve for $x$.

Step 4.1: Rearrange the equation to isolate $x$ on one side.

Step 4.2: Move all constants to the opposite side of the equation.

Step 4.2.1: Subtract $3$ from both sides to isolate $x$.

Step 4.2.2: Complete the subtraction to find the value of $x$.

Knowledge Notes:

1. Factoring: This involves rewriting an expression as a product of its factors. It is a crucial step in simplifying equations and expressions.

2. Least Common Denominator (LCD): The LCD is the smallest number that can be used as a common denominator for a set of fractions. It is typically found by identifying the least common multiple (LCM) of the denominators.

3. Least Common Multiple (LCM): The LCM of two or more numbers is the smallest number that is a multiple of all the numbers. It is found by listing the prime factors of each number and multiplying each factor by the greatest number of times it occurs in any of the numbers.

4. Prime Number: A prime number is a number greater than 1 that has no positive divisors other than 1 and itself.

5. Commutative Property of Multiplication: This property states that the order in which two numbers are multiplied does not change the product, i.e., $ab = ba$.

6. Distributive Property: This property allows one to distribute a factor across terms within parentheses, i.e., $a(b + c) = ab + ac$.

7. Solving Linear Equations: To solve a linear equation, one aims to isolate the variable on one side of the equation by performing operations that maintain the equality, such as adding, subtracting, multiplying, or dividing both sides by the same number.