Solve the Inequality for x 2/(x-1)>7/(x+1)

$\frac{2}{x - 1} > \frac{7}{x + 1}$

Subtract $\frac{7}{x + 1}$ from both sides of the inequality.

$\frac{2}{x - 1} - \frac{7}{x + 1} > 0$

Simplify $\frac{2}{x - 1} - \frac{7}{x + 1}$ .

To write $\frac{2}{x - 1}$ as a fraction with a common denominator, multiply by $\frac{x + 1}{x + 1}$ .

$\frac{2}{x - 1} \cdot \frac{x + 1}{x + 1} - \frac{7}{x + 1} > 0$

To write $- \frac{7}{x + 1}$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

$\frac{2}{x - 1} \cdot \frac{x + 1}{x + 1} - \frac{7}{x + 1} \cdot \frac{x - 1}{x - 1} > 0$

Write each expression with a common denominator of $(x - 1) (x + 1)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{2}{x - 1}$ by $\frac{x + 1}{x + 1}$ .

$\frac{2 (x + 1)}{(x - 1) (x + 1)} - \frac{7}{x + 1} \cdot \frac{x - 1}{x - 1} > 0$

Multiply $\frac{7}{x + 1}$ by $\frac{x - 1}{x - 1}$ .

$\frac{2 (x + 1)}{(x - 1) (x + 1)} - \frac{7 (x - 1)}{(x + 1) (x - 1)} > 0$

Reorder the factors of $(x - 1) (x + 1)$ .

$\frac{2 (x + 1)}{(x + 1) (x - 1)} - \frac{7 (x - 1)}{(x + 1) (x - 1)} > 0$

Combine the numerators over the common denominator.

$\frac{2 (x + 1) - 7 (x - 1)}{(x + 1) (x - 1)} > 0$

Simplify the numerator.

Apply the distributive property.

$\frac{2 x + 2 \cdot 1 - 7 (x - 1)}{(x + 1) (x - 1)} > 0$

Multiply $2$ by $1$ .

$\frac{2 x + 2 - 7 (x - 1)}{(x + 1) (x - 1)} > 0$

Apply the distributive property.

$\frac{2 x + 2 - 7 x - 7 \cdot - 1}{(x + 1) (x - 1)} > 0$

Multiply $- 7$ by $- 1$ .

$\frac{2 x + 2 - 7 x + 7}{(x + 1) (x - 1)} > 0$

Subtract $7 x$ from $2 x$ .

$\frac{- 5 x + 2 + 7}{(x + 1) (x - 1)} > 0$

Add $2$ and $7$ .

$\frac{- 5 x + 9}{(x + 1) (x - 1)} > 0$

Factor $- 1$ out of $- 5 x$ .

$\frac{- (5 x) + 9}{(x + 1) (x - 1)} > 0$

Rewrite $9$ as $- 1 (- 9)$ .

$\frac{- (5 x) - 1 (- 9)}{(x + 1) (x - 1)} > 0$

Factor $- 1$ out of $- (5 x) - 1 (- 9)$ .

$\frac{- (5 x - 9)}{(x + 1) (x - 1)} > 0$

Rewrite $- (5 x - 9)$ as $- 1 (5 x - 9)$ .

$\frac{- 1 (5 x - 9)}{(x + 1) (x - 1)} > 0$

Move the negative in front of the fraction.

$- \frac{5 x - 9}{(x + 1) (x - 1)} > 0$

Find all the values where the expression switches from negative to positive by setting each factor equal to $0$ and solving.

$5 x - 9 = 0$

$x + 1 = 0$

$x - 1 = 0$

Add $9$ to both sides of the equation.

$5 x = 9$

Divide each term in $5 x = 9$ by $5$ and simplify.

Divide each term in $5 x = 9$ by $5$ .

$\frac{5 x}{5} = \frac{9}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x}{5} = \frac{9}{5}$

Divide $x$ by $1$ .

$x = \frac{9}{5}$

Subtract $1$ from both sides of the equation.

$x = - 1$

Add $1$ to both sides of the equation.

$x = 1$

Solve for each factor to find the values where the absolute value expression goes from negative to positive.

$x = \frac{9}{5}$

$x = - 1$

$x = 1$

Consolidate the solutions.

$x = \frac{9}{5}, - 1, 1$

Find the domain of $- \frac{5 x - 9}{(x + 1) (x - 1)}$ .

Set the denominator in $\frac{5 x - 9}{(x + 1) (x - 1)}$ equal to $0$ to find where the expression is undefined.

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

Solve for $x$ .

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

$x + 1 = 0$

$x - 1 = 0$

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

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

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

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

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

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

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

$x = - 1, 1$

The domain is all values of $x$ that make the expression defined.

$(- \infty, - 1) \cup (- 1, 1) \cup (1, \infty)$

Use each root to create test intervals.

$x < - 1$

$- 1 < x < 1$

$1 < x < \frac{9}{5}$

$x > \frac{9}{5}$

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

Test a value on the interval $x < - 1$ to see if it makes the inequality true.

Choose a value on the interval $x < - 1$ and see if this value makes the original inequality true.

$x = - 4$

Replace $x$ with $- 4$ in the original inequality.

$\frac{2}{(- 4) - 1} > \frac{7}{(- 4) + 1}$

The left side $- 0.4$ is greater than the right side $- 2.\overline{3}$ , which means that the given statement is always true.

True

Test a value on the interval $- 1 < x < 1$ to see if it makes the inequality true.

Choose a value on the interval $- 1 < x < 1$ and see if this value makes the original inequality true.

$x = 0$

Replace $x$ with $0$ in the original inequality.

$\frac{2}{(0) - 1} > \frac{7}{(0) + 1}$

The left side $- 2$ is not greater than the right side $7$ , which means that the given statement is false.

False

Test a value on the interval $1 < x < \frac{9}{5}$ to see if it makes the inequality true.

Choose a value on the interval $1 < x < \frac{9}{5}$ and see if this value makes the original inequality true.

$x = 1.4$

Replace $x$ with $1.4$ in the original inequality.

$\frac{2}{(1.4) - 1} > \frac{7}{(1.4) + 1}$

The left side $5$ is greater than the right side $2.91\overline{6}$ , which means that the given statement is always true.

True

Test a value on the interval $x > \frac{9}{5}$ to see if it makes the inequality true.

Choose a value on the interval $x > \frac{9}{5}$ and see if this value makes the original inequality true.

$x = 4$

Replace $x$ with $4$ in the original inequality.

$\frac{2}{(4) - 1} > \frac{7}{(4) + 1}$

The left side $0.\overline{6}$ is not greater than the right side $1.4$ , which means that the given statement is false.

False

Compare the intervals to determine which ones satisfy the original inequality.

$x < - 1$ True

$- 1 < x < 1$ False

$1 < x < \frac{9}{5}$ True

$x > \frac{9}{5}$ False

$x < - 1$ True

$- 1 < x < 1$ False

$1 < x < \frac{9}{5}$ True

$x > \frac{9}{5}$ False

The solution consists of all of the true intervals.

$x < - 1$ or $1 < x < \frac{9}{5}$

The result can be shown in multiple forms.

Inequality Form:

$x < - 1 or 1 < x < \frac{9}{5}$

Interval Notation:

$(- \infty, - 1) \cup (1, \frac{9}{5})$

Solve the Inequality for x 2/(x-1)>7/(x+1)

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