Solve for x 1/5-(x-4)/10=1+(x-2)/2

Solution:

Step:1

Transform $\frac{1}{5} - \frac{x - 4}{10}$ to have a common denominator.

Step:1.1

Convert $\frac{1}{5}$ to have a denominator of $10$ by multiplying it by $\frac{2}{2}$: $\frac{1}{5} \cdot \frac{2}{2} - \frac{x - 4}{10} = 1 + \frac{x - 2}{2}$

Step:1.2

Ensure all terms have a denominator of $10$.

Step:1.2.1

Multiply $\frac{1}{5}$ by $\frac{2}{2}$ to get $\frac{2}{10} - \frac{x - 4}{10} = 1 + \frac{x - 2}{2}$

Step:1.2.2

Multiply $5$ by $2$ to confirm the common denominator: $\frac{2}{10} - \frac{x - 4}{10} = 1 + \frac{x - 2}{2}$

Step:1.3

Add the numerators over the shared denominator: $\frac{2 - (x - 4)}{10} = 1 + \frac{x - 2}{2}$

Step:1.4

Simplify the numerator by distributing.

Step:1.4.1

Distribute the negative sign: $\frac{2 - x + 4}{10} = 1 + \frac{x - 2}{2}$

Step:1.4.2

Combine like terms in the numerator: $\frac{6 - x}{10} = 1 + \frac{x - 2}{2}$

Step:1.5

Extract common factors if possible.

Step:1.5.1

Factor out $-1$ from $-x$: $\frac{-1(x - 6)}{10} = 1 + \frac{x - 2}{2}$

Step:1.5.2

Rewrite the expression: $- \frac{x - 6}{10} = 1 + \frac{x - 2}{2}$

Step:2

Simplify $1 + \frac{x - 2}{2}$.

Step:2.1

Combine into a single fraction.

Step:2.1.1

Express $1$ as a fraction with a denominator of $2$: $- \frac{x - 6}{10} = \frac{2}{2} + \frac{x - 2}{2}$

Step:2.1.2

Add the numerators together: $- \frac{x - 6}{10} = \frac{x}{2}$

Step:2.2

Simplify the numerator.

Step:2.2.1

Subtract $2$ from $2$: $- \frac{x - 6}{10} = \frac{x}{2}$

Step:3

Isolate all $x$ terms on one side.

Step:3.1

Subtract $\frac{x}{2}$ from both sides: $- \frac{x - 6}{10} - \frac{x}{2} = 0$

Step:3.2

Convert $- \frac{x}{2}$ to have a denominator of $10$ by multiplying by $\frac{5}{5}$: $- \frac{x - 6}{10} - \frac{x \cdot 5}{10} = 0$

Step:3.3

Combine the numerators over the common denominator: $\frac{- (x - 6) - 5x}{10} = 0$

Step:3.4

Simplify the numerator by distributing and combining like terms: $\frac{-6x + 6}{10} = 0$

Step:3.5

Factor out common terms if possible.

Step:3.5.1

Factor out $6$: $\frac{6(-x + 1)}{10} = 0$

Step:3.6

Reduce the fraction by canceling common factors.

Step:3.6.1

Divide both the numerator and the denominator by $2$: $\frac{3(-x + 1)}{5} = 0$

Step:3.7

Simplify further if possible: $- \frac{3(x - 1)}{5} = 0$

Step:4

Set the numerator equal to zero to solve for $x$: $3(x - 1) = 0$

Step:5

Find the value of $x$.

Step:5.1

Divide each term by $3$ and simplify.

Step:5.1.1

Divide the equation by $3$: $x - 1 = 0$

Step:5.2

Add $1$ to both sides to isolate $x$: $x = 1$

Knowledge Notes:

This problem involves solving a linear equation with fractions. Here are the relevant knowledge points:

1. Common Denominator: When adding or subtracting fractions, it's necessary to have a common denominator. This allows you to combine the fractions by adding or subtracting the numerators.

2. Distributive Property: This property is used to multiply a single term by each term inside a set of parentheses. For example, $a(b + c) = ab + ac$.

3. Combining Like Terms: This refers to the process of adding or subtracting terms that have the same variable raised to the same power.

4. Factoring: Factoring involves expressing an expression as the product of its factors. This can simplify the expression and is often used before canceling common factors.

5. Reducing Fractions: When a numerator and denominator share a common factor, they can be divided by that factor to simplify the fraction.

6. Solving Linear Equations: To solve for a variable, you need to isolate it on one side of the equation, often by using inverse operations like addition/subtraction or multiplication/division.