Algebra Examples

$2 \log (x) < \log (2 - x)$

Step 1

Convert the inequality to an equality.

$2 \log (x) = \log (2 - x)$

Step 2

Solve the equation.

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Step 2.1

Simplify $2 \log (x)$ by moving $2$ inside the logarithm.

$\log (x^{2}) = \log (2 - x)$

Step 2.2

For the equation to be equal, the argument of the logarithms on both sides of the equation must be equal.

$x^{2} = 2 - x$

Step 2.3

Solve for $x$ .

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Step 2.3.1

Add $x$ to both sides of the equation.

$x^{2} + x = 2$

Step 2.3.2

Subtract $2$ from both sides of the equation.

$x^{2} + x - 2 = 0$

Step 2.3.3

Factor $x^{2} + x - 2$ using the AC method.

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Step 2.3.3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 2$ and whose sum is $1$ .

$- 1, 2$

Step 2.3.3.2

Write the factored form using these integers.

$(x - 1) (x + 2) = 0$

Step 2.3.4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 1 = 0$

$x + 2 = 0$

Step 2.3.5

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 2.3.5.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 2.3.5.2

Add $1$ to both sides of the equation.

$x = 1$

Step 2.3.6

Set $x + 2$ equal to $0$ and solve for $x$ .

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Step 2.3.6.1

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 2.3.6.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 2.3.7

The final solution is all the values that make $(x - 1) (x + 2) = 0$ true.

$x = 1, - 2$

Step 3

Find the domain of $\log (\frac{x^{2}}{2 - x})$ .

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Step 3.1

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

$\frac{x^{2}}{2 - x} > 0$

Step 3.2

Solve for $x$ .

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Step 3.2.1

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

$x^{2} = 0$

$2 - x = 0$

Step 3.2.2

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

$x = \pm \sqrt{0}$

Step 3.2.3

Simplify $\pm \sqrt{0}$ .

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Step 3.2.3.1

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

$x = \pm \sqrt{0^{2}}$

Step 3.2.3.2

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

$x = \pm 0$

Step 3.2.3.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 3.2.4

Subtract $2$ from both sides of the equation.

$- x = - 2$

Step 3.2.5

Divide each term in $- x = - 2$ by $- 1$ and simplify.

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Step 3.2.5.1

Divide each term in $- x = - 2$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 2}{- 1}$

Step 3.2.5.2

Simplify the left side.

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Step 3.2.5.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 2}{- 1}$

Step 3.2.5.2.2

Divide $x$ by $1$ .

$x = \frac{- 2}{- 1}$

Step 3.2.5.3

Simplify the right side.

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Step 3.2.5.3.1

Divide $- 2$ by $- 1$ .

$x = 2$

Step 3.2.6

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

$x = 0$

$x = 2$

Step 3.2.7

Consolidate the solutions.

$x = 0, 2$

Step 3.2.8

Find the domain of $\frac{x^{2}}{2 - x}$ .

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Step 3.2.8.1

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

$2 - x = 0$

Step 3.2.8.2

Solve for $x$ .

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Step 3.2.8.2.1

Subtract $2$ from both sides of the equation.

$- x = - 2$

Step 3.2.8.2.2

Divide each term in $- x = - 2$ by $- 1$ and simplify.

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Step 3.2.8.2.2.1

Divide each term in $- x = - 2$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 2}{- 1}$

Step 3.2.8.2.2.2

Simplify the left side.

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Step 3.2.8.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 2}{- 1}$

Step 3.2.8.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 2}{- 1}$

Step 3.2.8.2.2.3

Simplify the right side.

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Step 3.2.8.2.2.3.1

Divide $- 2$ by $- 1$ .

$x = 2$

Step 3.2.8.3

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

$(- \infty, 2) \cup (2, \infty)$

Step 3.2.9

Use each root to create test intervals.

$x < 0$

$0 < x < 2$

$x > 2$

Step 3.2.10

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

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Step 3.2.10.1

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

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Step 3.2.10.1.1

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

$x = - 2$

Step 3.2.10.1.2

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

$\frac{{(- 2)}^{2}}{2 - (- 2)} > 0$

Step 3.2.10.1.3

The left side $1$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 3.2.10.2

Test a value on the interval $0 < x < 2$ to see if it makes the inequality true.

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Step 3.2.10.2.1

Choose a value on the interval $0 < x < 2$ and see if this value makes the original inequality true.

$x = 1$

Step 3.2.10.2.2

Replace $x$ with $1$ in the original inequality.

$\frac{{(1)}^{2}}{2 - (1)} > 0$

Step 3.2.10.2.3

The left side $1$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 3.2.10.3

Test a value on the interval $x > 2$ to see if it makes the inequality true.

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Step 3.2.10.3.1

Choose a value on the interval $x > 2$ and see if this value makes the original inequality true.

$x = 4$

Step 3.2.10.3.2

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

$\frac{{(4)}^{2}}{2 - (4)} > 0$

Step 3.2.10.3.3

The left side $- 8$ is not greater than the right side $0$ , which means that the given statement is false.

False

Step 3.2.10.4

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

$x < 0$ True

$0 < x < 2$ True

$x > 2$ False

$x < 0$ True

$0 < x < 2$ True

$x > 2$ False

Step 3.2.11

The solution consists of all of the true intervals.

$x < 0$ or $0 < x < 2$

Step 3.3

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

$2 - x = 0$

Step 3.4

Solve for $x$ .

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Step 3.4.1

Subtract $2$ from both sides of the equation.

$- x = - 2$

Step 3.4.2

Divide each term in $- x = - 2$ by $- 1$ and simplify.

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Step 3.4.2.1

Divide each term in $- x = - 2$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 2}{- 1}$

Step 3.4.2.2

Simplify the left side.

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Step 3.4.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 2}{- 1}$

Step 3.4.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 2}{- 1}$

Step 3.4.2.3

Simplify the right side.

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Step 3.4.2.3.1

Divide $- 2$ by $- 1$ .

$x = 2$

Step 3.5

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

$(- \infty, 0) \cup (0, 2)$

Step 4

Use each root to create test intervals.

$x < - 2$

$- 2 < x < 1$

$x > 1$

Step 5

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

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Step 5.1

Test a value on the interval $x < - 2$ to see if it makes the inequality true.

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Step 5.1.1

Choose a value on the interval $x < - 2$ and see if this value makes the original inequality true.

$x = - 4$

Step 5.1.2

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

$2 \log (- 4) < \log (2 - (- 4))$

Step 5.1.3

Determine if the inequality is true.

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Step 5.1.3.1

The equation cannot be solved because it is undefined.

$Undefined$

Step 5.1.3.2

The left side has no solution, which means that the given statement is false.

False

Step 5.2

Test a value on the interval $- 2 < x < 1$ to see if it makes the inequality true.

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Step 5.2.1

Choose a value on the interval $- 2 < x < 1$ and see if this value makes the original inequality true.

$x = 0$

Step 5.2.2

Replace $x$ with $0$ in the original inequality.

$2 \log (0) < \log (2 - (0))$

Step 5.2.3

Determine if the inequality is true.

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Step 5.2.3.1

The equation cannot be solved because it is undefined.

$Undefined$

Step 5.2.3.2

The left side has no solution, which means that the given statement is false.

False

Step 5.3

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

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Step 5.3.1

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

$x = 4$

Step 5.3.2

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

$2 \log (4) < \log (2 - (4))$

Step 5.3.3

Determine if the inequality is true.

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Step 5.3.3.1

The equation cannot be solved because it is undefined.

$Undefined$

Step 5.3.3.2

The right side has no solution, which means that the given statement is false.

False

Step 5.4

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

$x < - 2$ False

$- 2 < x < 1$ False

$x > 1$ False

$x < - 2$ False

$- 2 < x < 1$ False

$x > 1$ False

Step 6

Since there are no numbers that fall within the interval, this inequality has no solution.

No solution