Algebra Examples

$\frac{4 x}{x + 1} = \frac{(x + 3) (x - 1)}{x + 1}$

Step 1

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$4 x = (x + 3) (x - 1)$

Step 2

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$(x + 3) (x - 1) = 4 x$

Step 3

Simplify $(x + 3) (x - 1)$ .

Tap for more steps...

Step 3.1

Rewrite.

$0 + 0 + (x + 3) (x - 1) = 4 x$

Step 3.2

Simplify by adding zeros.

$(x + 3) (x - 1) = 4 x$

Step 3.3

Expand $(x + 3) (x - 1)$ using the FOIL Method.

Tap for more steps...

Step 3.3.1

Apply the distributive property.

$x (x - 1) + 3 (x - 1) = 4 x$

Step 3.3.2

Apply the distributive property.

$x \cdot x + x \cdot - 1 + 3 (x - 1) = 4 x$

Step 3.3.3

Apply the distributive property.

$x \cdot x + x \cdot - 1 + 3 x + 3 \cdot - 1 = 4 x$

Step 3.4

Simplify and combine like terms.

Tap for more steps...

Step 3.4.1

Simplify each term.

Tap for more steps...

Step 3.4.1.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 1 + 3 x + 3 \cdot - 1 = 4 x$

Step 3.4.1.2

Move $- 1$ to the left of $x$ .

$x^{2} - 1 \cdot x + 3 x + 3 \cdot - 1 = 4 x$

Step 3.4.1.3

Rewrite $- 1 x$ as $- x$ .

$x^{2} - x + 3 x + 3 \cdot - 1 = 4 x$

Step 3.4.1.4

Multiply $3$ by $- 1$ .

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

Step 3.4.2

Add $- x$ and $3 x$ .

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

Step 4

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

Tap for more steps...

Step 4.1

Subtract $4 x$ from both sides of the equation.

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

Step 4.2

Subtract $4 x$ from $2 x$ .

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

Step 5

Factor $x^{2} - 2 x - 3$ using the AC method.

Tap for more steps...

Step 5.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 3$ and whose sum is $- 2$ .

$- 3, 1$

Step 5.2

Write the factored form using these integers.

$(x - 3) (x + 1) = 0$

Step 6

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

$x - 3 = 0$

$x + 1 = 0$

Step 7

Set $x - 3$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 7.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 7.2

Add $3$ to both sides of the equation.

$x = 3$

Step 8

Set $x + 1$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 8.1

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 8.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 9

The final solution is all the values that make $(x - 3) (x + 1) = 0$ true.

$x = 3, - 1$

Step 10

Exclude the solutions that do not make $\frac{4 x}{x + 1} = \frac{(x + 3) (x - 1)}{x + 1}$ true.

$x = 3$