Linear Algebra Examples

$\log ({(2 x + 2)}^{2}) = \log (1 + \frac{12 x}{25}) + 2$

Step 1

Set the argument in $\log ({(2 x + 2)}^{2})$ greater than $0$ to find where the expression is defined.

${(2 x + 2)}^{2} > 0$

Step 2

Solve for $x$ .

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Step 2.1

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

$\sqrt{{(2 x + 2)}^{2}} > \sqrt{0}$

Step 2.2

Simplify the equation.

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Step 2.2.1

Simplify the left side.

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Step 2.2.1.1

Pull terms out from under the radical.

$| 2 x + 2 | > \sqrt{0}$

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

Simplify $\sqrt{0}$ .

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Step 2.2.2.1.1

Rewrite $0$ as $0^{2}$ .

$| 2 x + 2 | > \sqrt{0^{2}}$

Step 2.2.2.1.2

Pull terms out from under the radical.

$| 2 x + 2 | > | 0 |$

Step 2.2.2.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $0$ is $0$ .

$| 2 x + 2 | > 0$

Step 2.3

Write $| 2 x + 2 | > 0$ as a piecewise.

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Step 2.3.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$2 x + 2 \geq 0$

Step 2.3.2

Solve the inequality.

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Step 2.3.2.1

Subtract $2$ from both sides of the inequality.

$2 x \geq - 2$

Step 2.3.2.2

Divide each term in $2 x \geq - 2$ by $2$ and simplify.

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Step 2.3.2.2.1

Divide each term in $2 x \geq - 2$ by $2$ .

$\frac{2 x}{2} \geq \frac{- 2}{2}$

Step 2.3.2.2.2

Simplify the left side.

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Step 2.3.2.2.2.1

Cancel the common factor of $2$ .

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Step 2.3.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} \geq \frac{- 2}{2}$

Step 2.3.2.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{- 2}{2}$

Step 2.3.2.2.3

Simplify the right side.

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Step 2.3.2.2.3.1

Divide $- 2$ by $2$ .

$x \geq - 1$

Step 2.3.3

In the piece where $2 x + 2$ is non-negative, remove the absolute value.

$2 x + 2 > 0$

Step 2.3.4

To find the interval for the second piece, find where the inside of the absolute value is negative.

$2 x + 2 < 0$

Step 2.3.5

Solve the inequality.

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Step 2.3.5.1

Subtract $2$ from both sides of the inequality.

$2 x < - 2$

Step 2.3.5.2

Divide each term in $2 x < - 2$ by $2$ and simplify.

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Step 2.3.5.2.1

Divide each term in $2 x < - 2$ by $2$ .

$\frac{2 x}{2} < \frac{- 2}{2}$

Step 2.3.5.2.2

Simplify the left side.

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Step 2.3.5.2.2.1

Cancel the common factor of $2$ .

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Step 2.3.5.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} < \frac{- 2}{2}$

Step 2.3.5.2.2.1.2

Divide $x$ by $1$ .

$x < \frac{- 2}{2}$

Step 2.3.5.2.3

Simplify the right side.

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Step 2.3.5.2.3.1

Divide $- 2$ by $2$ .

$x < - 1$

Step 2.3.6

In the piece where $2 x + 2$ is negative, remove the absolute value and multiply by $- 1$ .

$- (2 x + 2) > 0$

Step 2.3.7

Write as a piecewise.

${\begin{cases} 2 x + 2 > 0 & x \geq - 1 \\ - (2 x + 2) > 0 & x < - 1 \end{cases}$

Step 2.3.8

Simplify $- (2 x + 2) > 0$ .

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Step 2.3.8.1

Apply the distributive property.

${\begin{cases} 2 x + 2 > 0 & x \geq - 1 \\ - (2 x) - 1 \cdot 2 > 0 & x < - 1 \end{cases}$

Step 2.3.8.2

Multiply $2$ by $- 1$ .

${\begin{cases} 2 x + 2 > 0 & x \geq - 1 \\ - 2 x - 1 \cdot 2 > 0 & x < - 1 \end{cases}$

Step 2.3.8.3

Multiply $- 1$ by $2$ .

${\begin{cases} 2 x + 2 > 0 & x \geq - 1 \\ - 2 x - 2 > 0 & x < - 1 \end{cases}$

Step 2.4

Solve $2 x + 2 > 0$ for $x$ .

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Step 2.4.1

Subtract $2$ from both sides of the inequality.

$2 x > - 2$

Step 2.4.2

Divide each term in $2 x > - 2$ by $2$ and simplify.

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Step 2.4.2.1

Divide each term in $2 x > - 2$ by $2$ .

$\frac{2 x}{2} > \frac{- 2}{2}$

Step 2.4.2.2

Simplify the left side.

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Step 2.4.2.2.1

Cancel the common factor of $2$ .

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Step 2.4.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} > \frac{- 2}{2}$

Step 2.4.2.2.1.2

Divide $x$ by $1$ .

$x > \frac{- 2}{2}$

Step 2.4.2.3

Simplify the right side.

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Step 2.4.2.3.1

Divide $- 2$ by $2$ .

$x > - 1$

Step 2.5

Solve $- 2 x - 2 > 0$ for $x$ .

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Step 2.5.1

Add $2$ to both sides of the inequality.

$- 2 x > 2$

Step 2.5.2

Divide each term in $- 2 x > 2$ by $- 2$ and simplify.

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Step 2.5.2.1

Divide each term in $- 2 x > 2$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 x}{- 2} < \frac{2}{- 2}$

Step 2.5.2.2

Simplify the left side.

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Step 2.5.2.2.1

Cancel the common factor of $- 2$ .

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Step 2.5.2.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} < \frac{2}{- 2}$

Step 2.5.2.2.1.2

Divide $x$ by $1$ .

$x < \frac{2}{- 2}$

Step 2.5.2.3

Simplify the right side.

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Step 2.5.2.3.1

Divide $2$ by $- 2$ .

$x < - 1$

Step 2.6

Find the union of the solutions.

$x < - 1$ or $x > - 1$

Step 3

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

Interval Notation:

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

Set-Builder Notation:

${x | x \neq - 1}$

Step 4