Solve for x (x-4)/6-2=x/2

Solution:

Step 1

Scale up both sides by multiplying by $2$.

$2 \left( \frac{x - 4}{6} - 2 \right) = 2 \left( \frac{x}{2} \right)$

Step 2

Begin simplification process.

Step 2.1

Focus on the left-hand side first.

Step 2.1.1

Work on $2 \left( \frac{x - 4}{6} - 2 \right)$.

Step 2.1.1.1

Convert $-2$ to a fraction with the same denominator by multiplying by $\frac{6}{6}$.

$2 \left( \frac{x - 4}{6} - 2 \cdot \frac{6}{6} \right) = 2 \left( \frac{x}{2} \right)$

Step 2.1.1.2

Merge terms into a single fraction.

$2 \left( \frac{x - 4 - 2 \cdot 6}{6} \right) = 2 \left( \frac{x}{2} \right)$

Step 2.1.1.3

Condense the numerator by performing the arithmetic.

$2 \left( \frac{x - 16}{6} \right) = 2 \left( \frac{x}{2} \right)$

Step 2.1.1.4

Eliminate the factor of $2$.

$2 \cdot \frac{x - 16}{3 \cdot 2} = 2 \cdot \frac{x}{2}$

Step 2.1.1.4.1

Rewrite the simplified expression.

$\frac{x - 16}{3} = x$

Step 2.2

Now, address the right-hand side.

Step 2.2.1

Remove the common factor of $2$.

$\frac{x - 16}{3} = \frac{x}{2} \cdot 2$

Step 2.2.1.1

Rewrite the simplified expression.

$\frac{x - 16}{3} = x$

Step 3

Isolate $x$ to find its value.

Step 3.1

Multiply both sides by $3$.

$3 \cdot \frac{x - 16}{3} = 3 \cdot x$

Step 3.2

Simplify both sides.

Step 3.2.1

Eliminate the common factor of $3$ on the left.

$x - 16 = 3 \cdot x$

Step 3.2.2

Arrange the equation to have all $x$ terms on one side.

$x - 16 = 3x$

Step 3.3

Final steps to solve for $x$.

Step 3.3.1

Consolidate $x$ terms on one side.

$x - 3x = 16$

Step 3.3.2

Combine like terms.

$-2x = 16$

Step 3.3.3

Divide to solve for $x$.

$x = \frac{16}{-2}$

Step 3.3.3.1

Complete the division.

$x = -8$

Knowledge Notes:

1. Multiplication of Fractions: To multiply a fraction by a whole number, you can multiply the numerator by the whole number while keeping the denominator the same.

2. Common Denominator: To combine fractions, they must have the same denominator. You can convert whole numbers to fractions with a common denominator by multiplying by a fraction equivalent to 1 (e.g., $\frac{6}{6}$).

3. Simplifying Expressions: Combining like terms and canceling common factors are standard techniques used to simplify algebraic expressions.

4. Solving Linear Equations: To solve for a variable, you need to isolate it on one side of the equation. This often involves moving terms from one side to the other by performing inverse operations (e.g., adding or subtracting terms, multiplying or dividing by coefficients).

5. Inverse Operations: These are operations that reverse the effect of another operation. For example, multiplication is the inverse of division, and addition is the inverse of subtraction. They are used to isolate variables in equations.

6. Cross Multiplication: When you have an equation with fractions on both sides, you can cross multiply to eliminate the denominators and solve for the variable.