Solve the Rational Equation for x (5x)/(x+2)=(3x)/(x+1)

$\frac{5 x}{x + 2} = \frac{3 x}{x + 1}$

Step 1

Multiply the numerator of the first fraction by the denominator of the second fraction. Set this equal to the product of the denominator of the first fraction and the numerator of the second fraction.

$5 x (x + 1) = (x + 2) (3 x)$

Step 2

Solve the equation for $x$ .

Simplify $5 x (x + 1)$ .

Start simplifying.

$0 + 0 + 5 x (x + 1) = (x + 2) \cdot (3 x)$

Simplify by adding zeros.

$5 x (x + 1) = (x + 2) \cdot (3 x)$

Apply the distributive property.

$5 x \cdot x + 5 x \cdot 1 = (x + 2) \cdot (3 x)$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$5 (x \cdot x) + 5 x \cdot 1 = (x + 2) \cdot (3 x)$

Multiply $x$ by $x$ .

$5 x^{2} + 5 x \cdot 1 = (x + 2) \cdot (3 x)$

Multiply $5$ by $1$ .

$5 x^{2} + 5 x = (x + 2) \cdot (3 x)$

Simplify $(x + 2) \cdot (3 x)$ .

Apply the distributive property.

$5 x^{2} + 5 x = x (3 x) + 2 (3 x)$

Simplify the expression.

Rewrite using the commutative property of multiplication.

$5 x^{2} + 5 x = 3 x \cdot x + 2 (3 x)$

Multiply $3$ by $2$ .

$5 x^{2} + 5 x = 3 x \cdot x + 6 x$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$5 x^{2} + 5 x = 3 (x \cdot x) + 6 x$

Multiply $x$ by $x$ .

$5 x^{2} + 5 x = 3 x^{2} + 6 x$

Move all terms containing $x$ to the left side of the equation.

Subtract $3 x^{2}$ from both sides of the equation.

$5 x^{2} + 5 x - 3 x^{2} = 6 x$

Subtract $6 x$ from both sides of the equation.

$5 x^{2} + 5 x - 3 x^{2} - 6 x = 0$

Subtract $3 x^{2}$ from $5 x^{2}$ .

$2 x^{2} + 5 x - 6 x = 0$

Subtract $6 x$ from $5 x$ .

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

Factor $x$ out of $2 x^{2} - x$ .

Factor $x$ out of $2 x^{2}$ .

$x (2 x) - x = 0$

Factor $x$ out of $- x$ .

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

Factor $x$ out of $x (2 x) + x \cdot - 1$ .

$x (2 x - 1) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$2 x - 1 = 0$

Set $x$ equal to $0$ .

$x = 0$

Set $2 x - 1$ equal to $0$ and solve for $x$ .

Set $2 x - 1$ equal to $0$ .

$2 x - 1 = 0$

Solve $2 x - 1 = 0$ for $x$ .

Add $1$ to both sides of the equation.

$2 x = 1$

Divide each term in $2 x = 1$ by $2$ and simplify.

Divide each term in $2 x = 1$ by $2$ .

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

The final solution is all the values that make $x (2 x - 1) = 0$ true.

$x = 0, \frac{1}{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = 0, \frac{1}{2}$

Decimal Form:

$x = 0, 0.5$

Solve the Rational Equation for x (5x)/(x+2)=(3x)/(x+1)

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