Solve for x (x+2)/x=(x-1)/2

The problem presented is a rational equation where the variable x appears in both the numerator and denominator of two separate fractions. The equation equates the quotient of x added to two, all over x, to the quotient of x subtracted by one, all over two. To solve for x, one would typically find a common denominator to clear the fractions and then solve the resulting linear equation.

$\frac{x + 2}{x} = \frac{x - 1}{2}$

Answer

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Solution:

Step:1

Cross-multiply to eliminate the fractions. Multiply $(x + 2)$ by $2$ and set it equal to $x$ multiplied by $(x - 1)$ . $(x + 2) \cdot 2 = x \cdot (x - 1)$

Step:2

Begin solving for $x$ .

Step:2.1

Rearrange the equation to have $x$ terms on the left side.

$x \cdot (x - 1) = (x + 2) \cdot 2$

Step:2.2

Expand $x \cdot (x - 1)$ .

Step:2.2.1

Introduce a zero addition to the equation.

$0 + 0 + x \cdot (x - 1) = (x + 2) \cdot 2$

Step:2.2.2

Distribute $x$ across $(x - 1)$ .

Step:2.2.2.1

Use the distributive property to expand.

$x \cdot x - x \cdot 1 = (x + 2) \cdot 2$

Step:2.2.2.2

Combine like terms.

$x^{2} - x = (x + 2) \cdot 2$

Step:2.2.3

Rewrite $- 1 \cdot x$ as $- x$ .

$x^{2} - x = (x + 2) \cdot 2$

Step:2.3

Expand $(x + 2) \cdot 2$ .

Step:2.3.1

Apply the distributive property.

$x^{2} - x = 2 \cdot x + 2 \cdot 2$

Step:2.3.2

Simplify the right side of the equation.

$x^{2} - x = 2 x + 4$

Step:2.4

Consolidate all $x$ terms on one side.

Step:2.4.1

Subtract $2 x$ from both sides.

$x^{2} - x - 2 x = 4$

Step:2.4.2

Combine the $x$ terms.

$x^{2} - 3 x = 4$

Step:2.5

Get all terms on one side to set the equation to zero.

$x^{2} - 3 x - 4 = 0$

Step:2.6

Factor the quadratic equation.

Step:2.6.1

Identify two numbers that multiply to $- 4$ and add up to $- 3$ .

$- 4, 1$

Step:2.6.2

Write the factors based on these numbers.

$(x - 4) (x + 1) = 0$

Step:2.7

Apply the zero-product property.

$x - 4 = 0 or x + 1 = 0$

Step:2.8

Solve $x - 4 = 0$ for $x$ .

Step:2.8.1

Set up the equation.

$x - 4 = 0$

Step:2.8.2

Add $4$ to isolate $x$ .

$x = 4$

Step:2.9

Solve $x + 1 = 0$ for $x$ .

Step:2.9.1

Set up the equation.

$x + 1 = 0$

Step:2.9.2

Subtract $1$ to isolate $x$ .

$x = - 1$

Step:2.10

Combine the solutions to find the values of $x$ .

$x = 4 or x = - 1$

Knowledge Notes:

To solve the given equation $(x + 2) / x = (x - 1) / 2$ , we use the cross-multiplication method to eliminate the fractions. This is done by multiplying the numerator of one fraction by the denominator of the other and setting the products equal to each other. After cross-multiplication, we rearrange the terms to isolate $x$ on one side of the equation.

The distributive property, $a (b + c) = a b + a c$ , is used to expand expressions such as $x (x - 1)$ and $(x + 2) \cdot 2$ . It allows us to simplify the equation by distributing a single term across terms within parentheses.

Combining like terms is a process where we add or subtract terms with the same variable raised to the same power. For example, $- x - 2 x$ combines to $- 3 x$ .

The zero-product property states that if the product of two factors is zero, then at least one of the factors must be zero. This property is used to solve quadratic equations that have been factored into the form $(x - a) (x - b) = 0$ .

Factoring is the process of breaking down an expression into a product of simpler expressions. In the case of a quadratic equation, we look for two numbers that multiply to the constant term and add to the coefficient of the $x$ term. Once factored, we use the zero-product property to find the solutions for $x$ .

The final solution to the equation is the set of all values of $x$ that satisfy the original equation, which in this case are $x = 4$ and $x = - 1$ .