Precalculus Examples

$x^{2} + 6 x > - 10$

Step 1

Add $10$ to both sides of the inequality.

$x^{2} + 6 x + 10 > 0$

Step 2

Convert the inequality to an equation.

$x^{2} + 6 x + 10 = 0$

Step 3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4

Substitute the values $a = 1$ , $b = 6$ , and $c = 10$ into the quadratic formula and solve for $x$ .

$\frac{- 6 \pm \sqrt{6^{2} - 4 \cdot (1 \cdot 10)}}{2 \cdot 1}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $6$ to the power of $2$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot 1 \cdot 10}}{2 \cdot 1}$

Step 5.1.2

Multiply $- 4 \cdot 1 \cdot 10$ .

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Step 5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot 10}}{2 \cdot 1}$

Step 5.1.2.2

Multiply $- 4$ by $10$ .

$x = \frac{- 6 \pm \sqrt{36 - 40}}{2 \cdot 1}$

Step 5.1.3

Subtract $40$ from $36$ .

$x = \frac{- 6 \pm \sqrt{- 4}}{2 \cdot 1}$

Step 5.1.4

Rewrite $- 4$ as $- 1 (4)$ .

$x = \frac{- 6 \pm \sqrt{- 1 \cdot 4}}{2 \cdot 1}$

Step 5.1.5

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$x = \frac{- 6 \pm \sqrt{- 1} \cdot \sqrt{4}}{2 \cdot 1}$

Step 5.1.6

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

$x = \frac{- 6 \pm i \cdot \sqrt{4}}{2 \cdot 1}$

Step 5.1.7

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

$x = \frac{- 6 \pm i \cdot \sqrt{2^{2}}}{2 \cdot 1}$

Step 5.1.8

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{- 6 \pm i \cdot 2}{2 \cdot 1}$

Step 5.1.9

Move $2$ to the left of $i$ .

$x = \frac{- 6 \pm 2 i}{2 \cdot 1}$

Step 5.2

Multiply $2$ by $1$ .

$x = \frac{- 6 \pm 2 i}{2}$

Step 5.3

Simplify $\frac{- 6 \pm 2 i}{2}$ .

$x = - 3 \pm i$

Step 6

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $6$ to the power of $2$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot 1 \cdot 10}}{2 \cdot 1}$

Step 6.1.2

Multiply $- 4 \cdot 1 \cdot 10$ .

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Step 6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot 10}}{2 \cdot 1}$

Step 6.1.2.2

Multiply $- 4$ by $10$ .

$x = \frac{- 6 \pm \sqrt{36 - 40}}{2 \cdot 1}$

Step 6.1.3

Subtract $40$ from $36$ .

$x = \frac{- 6 \pm \sqrt{- 4}}{2 \cdot 1}$

Step 6.1.4

Rewrite $- 4$ as $- 1 (4)$ .

$x = \frac{- 6 \pm \sqrt{- 1 \cdot 4}}{2 \cdot 1}$

Step 6.1.5

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$x = \frac{- 6 \pm \sqrt{- 1} \cdot \sqrt{4}}{2 \cdot 1}$

Step 6.1.6

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

$x = \frac{- 6 \pm i \cdot \sqrt{4}}{2 \cdot 1}$

Step 6.1.7

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

$x = \frac{- 6 \pm i \cdot \sqrt{2^{2}}}{2 \cdot 1}$

Step 6.1.8

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{- 6 \pm i \cdot 2}{2 \cdot 1}$

Step 6.1.9

Move $2$ to the left of $i$ .

$x = \frac{- 6 \pm 2 i}{2 \cdot 1}$

Step 6.2

Multiply $2$ by $1$ .

$x = \frac{- 6 \pm 2 i}{2}$

Step 6.3

Simplify $\frac{- 6 \pm 2 i}{2}$ .

$x = - 3 \pm i$

Step 6.4

Change the $\pm$ to $+$ .

$x = - 3 + i$

Step 7

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $6$ to the power of $2$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot 1 \cdot 10}}{2 \cdot 1}$

Step 7.1.2

Multiply $- 4 \cdot 1 \cdot 10$ .

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Step 7.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot 10}}{2 \cdot 1}$

Step 7.1.2.2

Multiply $- 4$ by $10$ .

$x = \frac{- 6 \pm \sqrt{36 - 40}}{2 \cdot 1}$

Step 7.1.3

Subtract $40$ from $36$ .

$x = \frac{- 6 \pm \sqrt{- 4}}{2 \cdot 1}$

Step 7.1.4

Rewrite $- 4$ as $- 1 (4)$ .

$x = \frac{- 6 \pm \sqrt{- 1 \cdot 4}}{2 \cdot 1}$

Step 7.1.5

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$x = \frac{- 6 \pm \sqrt{- 1} \cdot \sqrt{4}}{2 \cdot 1}$

Step 7.1.6

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

$x = \frac{- 6 \pm i \cdot \sqrt{4}}{2 \cdot 1}$

Step 7.1.7

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

$x = \frac{- 6 \pm i \cdot \sqrt{2^{2}}}{2 \cdot 1}$

Step 7.1.8

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{- 6 \pm i \cdot 2}{2 \cdot 1}$

Step 7.1.9

Move $2$ to the left of $i$ .

$x = \frac{- 6 \pm 2 i}{2 \cdot 1}$

Step 7.2

Multiply $2$ by $1$ .

$x = \frac{- 6 \pm 2 i}{2}$

Step 7.3

Simplify $\frac{- 6 \pm 2 i}{2}$ .

$x = - 3 \pm i$

Step 7.4

Change the $\pm$ to $-$ .

$x = - 3 - i$

Step 8

Identify the leading coefficient.

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Step 8.1

The leading term in a polynomial is the term with the highest degree.

$x^{2}$

Step 8.2

The leading coefficient in a polynomial is the coefficient of the leading term.

$1$

Step 9

Since there are no real x-intercepts and the leading coefficient is positive, the parabola opens up and $x^{2} + 6 x$ is always greater than $0$ .

All real numbers

Step 10

Convert the inequality to interval notation.

$(- \infty, \infty)$

Step 11