Solve the Inequality for x 7+2/x<-5/x

$7 + \frac{2}{x} < \frac{- 5}{x}$

Multiply both sides by $x$ .

$(7 + \frac{2}{x}) x = \frac{- 5}{x} x$

Simplify.

Simplify the left side.

Simplify $(7 + \frac{2}{x}) x$ .

Apply the distributive property.

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

Cancel the common factor of $x$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify the right side.

Cancel the common factor of $x$ .

Cancel the common factor.

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

Rewrite the expression.

$7 x + 2 = - 5$

Solve for $x$ .

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

Subtract $2$ from both sides of the equation.

$7 x = - 5 - 2$

Subtract $2$ from $- 5$ .

$7 x = - 7$

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

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

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

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $- 7$ by $7$ .

$x = - 1$

Find the domain of $7 + \frac{7}{x}$ .

Set the denominator in $\frac{7}{x}$ equal to $0$ to find where the expression is undefined.

$x = 0$

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

$(- \infty, 0) \cup (0, \infty)$

Use each root to create test intervals.

$x < - 1$

$- 1 < x < 0$

$x > 0$

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.

$7 + \frac{2}{- 4} < \frac{- 5}{- 4}$

The left side $6.5$ is not less than the right side $1.25$ , which means that the given statement is false.

False

Test a value on the interval $- 1 < x < 0$ to see if it makes the inequality true.

Choose a value on the interval $- 1 < x < 0$ and see if this value makes the original inequality true.

$x = - 0.5$

Replace $x$ with $- 0.5$ in the original inequality.

$7 + \frac{2}{- 0.5} < \frac{- 5}{- 0.5}$

The left side $3$ is less than the right side $10$ , which means that the given statement is always true.

True

Test a value on the interval $x > 0$ to see if it makes the inequality true.

Choose a value on the interval $x > 0$ and see if this value makes the original inequality true.

$x = 2$

Replace $x$ with $2$ in the original inequality.

$7 + \frac{2}{2} < \frac{- 5}{2}$

The left side $8$ is not less than the right side $- 2.5$ , which means that the given statement is false.

False

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

$x < - 1$ False

$- 1 < x < 0$ True

$x > 0$ False

$x < - 1$ False

$- 1 < x < 0$ True

$x > 0$ False

The solution consists of all of the true intervals.

$- 1 < x < 0$

The result can be shown in multiple forms.

Inequality Form:

$- 1 < x < 0$

Interval Notation:

$(- 1, 0)$

Solve the Inequality for x 7+2/x<-5/x

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