Algebra Examples

$\frac{x^{2} + 1}{5 x + 11} \leq 0$

Step 1

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

$x^{2} + 1 = 0$

$5 x + 11 = 0$

Step 2

Subtract $1$ from both sides of the equation.

$x^{2} = - 1$

Step 3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{- 1}$

Step 4

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i$

Step 5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 5.1

First, use the positive value of the $\pm$ to find the first solution.

$x = i$

Step 5.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - i$

Step 5.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = i, - i$

Step 6

Subtract $11$ from both sides of the equation.

$5 x = - 11$

Step 7

Divide each term in $5 x = - 11$ by $5$ and simplify.

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Step 7.1

Divide each term in $5 x = - 11$ by $5$ .

$\frac{5 x}{5} = \frac{- 11}{5}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $5$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{- 11}{5}$

Step 7.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 11}{5}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Move the negative in front of the fraction.

$x = - \frac{11}{5}$

Step 8

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

$x = i, - i$

$x = - \frac{11}{5}$

Step 9

Find the domain of $\frac{x^{2} + 1}{5 x + 11}$ .

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Step 9.1

Set the denominator in $\frac{x^{2} + 1}{5 x + 11}$ equal to $0$ to find where the expression is undefined.

$5 x + 11 = 0$

Step 9.2

Solve for $x$ .

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Step 9.2.1

Subtract $11$ from both sides of the equation.

$5 x = - 11$

Step 9.2.2

Divide each term in $5 x = - 11$ by $5$ and simplify.

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Step 9.2.2.1

Divide each term in $5 x = - 11$ by $5$ .

$\frac{5 x}{5} = \frac{- 11}{5}$

Step 9.2.2.2

Simplify the left side.

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Step 9.2.2.2.1

Cancel the common factor of $5$ .

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Step 9.2.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{- 11}{5}$

Step 9.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 11}{5}$

Step 9.2.2.3

Simplify the right side.

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Step 9.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{11}{5}$

Step 9.3

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

$(- \infty, - \frac{11}{5}) \cup (- \frac{11}{5}, \infty)$

Step 10

The solution consists of all of the true intervals.

$x < - \frac{11}{5}$

Step 11

The result can be shown in multiple forms.

Inequality Form:

$x < - \frac{11}{5}$

Interval Notation:

$(- \infty, - \frac{11}{5})$

Step 12