Algebra Examples

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

Step 1

Multiply both sides by $x$ .

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

Step 2

Simplify.

Tap for more steps...

Step 2.1

Simplify the left side.

Tap for more steps...

Step 2.1.1

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

Tap for more steps...

Step 2.1.1.1

Apply the distributive property.

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

Step 2.1.1.2

Cancel the common factor of $x$ .

Tap for more steps...

Step 2.1.1.2.1

Cancel the common factor.

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

Step 2.1.1.2.2

Rewrite the expression.

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

Step 2.2

Simplify the right side.

Tap for more steps...

Step 2.2.1

Cancel the common factor of $x$ .

Tap for more steps...

Step 2.2.1.1

Cancel the common factor.

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

Step 2.2.1.2

Rewrite the expression.

$7 x + 2 = - 5$

Step 3

Solve for $x$ .

Tap for more steps...

Step 3.1

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

Tap for more steps...

Step 3.1.1

Subtract $2$ from both sides of the equation.

$7 x = - 5 - 2$

Step 3.1.2

Subtract $2$ from $- 5$ .

$7 x = - 7$

Step 3.2

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

Tap for more steps...

Step 3.2.1

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

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

Step 3.2.2

Simplify the left side.

Tap for more steps...

Step 3.2.2.1

Cancel the common factor of $7$ .

Tap for more steps...

Step 3.2.2.1.1

Cancel the common factor.

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

Step 3.2.2.1.2

Divide $x$ by $1$ .

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

Step 3.2.3

Simplify the right side.

Tap for more steps...

Step 3.2.3.1

Divide $- 7$ by $7$ .

$x = - 1$

Step 4

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

Tap for more steps...

Step 4.1

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

$x = 0$

Step 4.2

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

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

Step 5

Use each root to create test intervals.

$x < - 1$

$- 1 < x < 0$

$x > 0$

Step 6

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

Tap for more steps...

Step 6.1

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

Tap for more steps...

Step 6.1.1

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

$x = - 4$

Step 6.1.2

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

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

Step 6.1.3

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

False

Step 6.2

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

Tap for more steps...

Step 6.2.1

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

$x = - 0.5$

Step 6.2.2

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

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

Step 6.2.3

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

True

Step 6.3

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

Tap for more steps...

Step 6.3.1

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

$x = 2$

Step 6.3.2

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

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

Step 6.3.3

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

False

Step 6.4

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

Step 7

The solution consists of all of the true intervals.

$- 1 < x < 0$

Step 8

The result can be shown in multiple forms.

Inequality Form:

$- 1 < x < 0$

Interval Notation:

$(- 1, 0)$

Step 9