Precalculus Examples

$\frac{2 x}{x + 7} = \frac{5}{x + 1}$

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.

$2 x (x + 1) = (x + 7) \cdot 5$

Step 2

Solve the equation for $x$ .

Tap for more steps...

Step 2.1

Simplify $2 x (x + 1)$ .

Tap for more steps...

Step 2.1.1

Rewrite.

$0 + 0 + 2 x (x + 1) = (x + 7) \cdot 5$

Step 2.1.2

Simplify by adding zeros.

$2 x (x + 1) = (x + 7) \cdot 5$

Step 2.1.3

Apply the distributive property.

$2 x \cdot x + 2 x \cdot 1 = (x + 7) \cdot 5$

Step 2.1.4

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

Tap for more steps...

Step 2.1.4.1

Move $x$ .

$2 (x \cdot x) + 2 x \cdot 1 = (x + 7) \cdot 5$

Step 2.1.4.2

Multiply $x$ by $x$ .

$2 x^{2} + 2 x \cdot 1 = (x + 7) \cdot 5$

Step 2.1.5

Multiply $2$ by $1$ .

$2 x^{2} + 2 x = (x + 7) \cdot 5$

Step 2.2

Simplify $(x + 7) \cdot 5$ .

Tap for more steps...

Step 2.2.1

Apply the distributive property.

$2 x^{2} + 2 x = x \cdot 5 + 7 \cdot 5$

Step 2.2.2

Simplify the expression.

Tap for more steps...

Step 2.2.2.1

Move $5$ to the left of $x$ .

$2 x^{2} + 2 x = 5 \cdot x + 7 \cdot 5$

Step 2.2.2.2

Multiply $7$ by $5$ .

$2 x^{2} + 2 x = 5 x + 35$

Step 2.3

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

Tap for more steps...

Step 2.3.1

Subtract $5 x$ from both sides of the equation.

$2 x^{2} + 2 x - 5 x = 35$

Step 2.3.2

Subtract $5 x$ from $2 x$ .

$2 x^{2} - 3 x = 35$

Step 2.4

Subtract $35$ from both sides of the equation.

$2 x^{2} - 3 x - 35 = 0$

Step 2.5

Factor by grouping.

Tap for more steps...

Step 2.5.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 35 = - 70$ and whose sum is $b = - 3$ .

Tap for more steps...

Step 2.5.1.1

Factor $- 3$ out of $- 3 x$ .

$2 x^{2} - 3 x - 35 = 0$

Step 2.5.1.2

Rewrite $- 3$ as $7$ plus $- 10$

$2 x^{2} + (7 - 10) x - 35 = 0$

Step 2.5.1.3

Apply the distributive property.

$2 x^{2} + 7 x - 10 x - 35 = 0$

Step 2.5.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 2.5.2.1

Group the first two terms and the last two terms.

$(2 x^{2} + 7 x) - 10 x - 35 = 0$

Step 2.5.2.2

Factor out the greatest common factor (GCF) from each group.

$x (2 x + 7) - 5 (2 x + 7) = 0$

Step 2.5.3

Factor the polynomial by factoring out the greatest common factor, $2 x + 7$ .

$(2 x + 7) (x - 5) = 0$

Step 2.6

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

$2 x + 7 = 0$

$x - 5 = 0$

Step 2.7

Set $2 x + 7$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 2.7.1

Set $2 x + 7$ equal to $0$ .

$2 x + 7 = 0$

Step 2.7.2

Solve $2 x + 7 = 0$ for $x$ .

Tap for more steps...

Step 2.7.2.1

Subtract $7$ from both sides of the equation.

$2 x = - 7$

Step 2.7.2.2

Divide each term in $2 x = - 7$ by $2$ and simplify.

Tap for more steps...

Step 2.7.2.2.1

Divide each term in $2 x = - 7$ by $2$ .

$\frac{2 x}{2} = \frac{- 7}{2}$

Step 2.7.2.2.2

Simplify the left side.

Tap for more steps...

Step 2.7.2.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.7.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 7}{2}$

Step 2.7.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 7}{2}$

Step 2.7.2.2.3

Simplify the right side.

Tap for more steps...

Step 2.7.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{7}{2}$

Step 2.8

Set $x - 5$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 2.8.1

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Step 2.8.2

Add $5$ to both sides of the equation.

$x = 5$

Step 2.9

The final solution is all the values that make $(2 x + 7) (x - 5) = 0$ true.

$x = - \frac{7}{2}, 5$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{7}{2}, 5$

Decimal Form:

$x = - 3.5, 5$

Mixed Number Form:

$x = - 3 \frac{1}{2}, 5$