Solve for x (x+2)/(x^2-4)+11=8

$\frac{x + 2}{x^{2} - 4} + 11 = 8$

Step 1

Factor each term.

Rewrite $4$ as $2^{2}$ .

$\frac{x + 2}{x^{2} - 2^{2}} + 11 = 8$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

$\frac{x + 2}{(x + 2) (x - 2)} + 11 = 8$

Reduce the expression $\frac{x + 2}{(x + 2) (x - 2)}$ by cancelling the common factors.

Cancel the common factor.

$\frac{x + 2}{(x + 2) (x - 2)} + 11 = 8$

Rewrite the expression.

$\frac{1}{x - 2} + 11 = 8$

Step 2

Find the LCD of the terms in the equation.

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x - 2, 1, 1$

Remove parentheses.

$x - 2, 1, 1$

The LCM of one and any expression is the expression.

$x - 2$

Step 3

Multiply each term in $\frac{1}{x - 2} + 11 = 8$ by $x - 2$ to eliminate the fractions.

Multiply each term in $\frac{1}{x - 2} + 11 = 8$ by $x - 2$ .

$\frac{1}{x - 2} (x - 2) + 11 (x - 2) = 8 (x - 2)$

Simplify the left side.

Simplify each term.

Cancel the common factor of $x - 2$ .

Cancel the common factor.

$\frac{1}{x - 2} (x - 2) + 11 (x - 2) = 8 (x - 2)$

Rewrite the expression.

$1 + 11 (x - 2) = 8 (x - 2)$

Apply the distributive property.

$1 + 11 x + 11 \cdot - 2 = 8 (x - 2)$

Multiply $11$ by $- 2$ .

$1 + 11 x - 22 = 8 (x - 2)$

Subtract $22$ from $1$ .

$11 x - 21 = 8 (x - 2)$

Simplify the right side.

Apply the distributive property.

$11 x - 21 = 8 x + 8 \cdot - 2$

Multiply $8$ by $- 2$ .

$11 x - 21 = 8 x - 16$

Step 4

Solve the equation.

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

Subtract $8 x$ from both sides of the equation.

$11 x - 21 - 8 x = - 16$

Subtract $8 x$ from $11 x$ .

$3 x - 21 = - 16$

Move all terms not containing $x$ to the right side of the equation.

Add $21$ to both sides of the equation.

$3 x = - 16 + 21$

Add $- 16$ and $21$ .

$3 x = 5$

Divide each term in $3 x = 5$ by $3$ and simplify.

Divide each term in $3 x = 5$ by $3$ .

$\frac{3 x}{3} = \frac{5}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{5}{3}$

Divide $x$ by $1$ .

$x = \frac{5}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5}{3}$

Decimal Form:

$x = 1.\overline{6}$

Mixed Number Form:

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

Solve for x (x+2)/(x^2-4)+11=8

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