Linear Algebra Examples

$y = \sqrt{\ln (\frac{4 - x}{x - 2})}$

Step 1

Set the argument in $\ln (\frac{4 - x}{x - 2})$ greater than $0$ to find where the expression is defined.

$\frac{4 - x}{x - 2} > 0$

Step 2

Solve for $x$ .

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Step 2.1

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

$4 - x = 0$

$x - 2 = 0$

Step 2.2

Subtract $4$ from both sides of the equation.

$- x = - 4$

Step 2.3

Divide each term in $- x = - 4$ by $- 1$ and simplify.

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Step 2.3.1

Divide each term in $- x = - 4$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 4}{- 1}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 4}{- 1}$

Step 2.3.2.2

Divide $x$ by $1$ .

$x = \frac{- 4}{- 1}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Divide $- 4$ by $- 1$ .

$x = 4$

Step 2.4

Add $2$ to both sides of the equation.

$x = 2$

Step 2.5

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

$x = 4$

$x = 2$

Step 2.6

Consolidate the solutions.

$x = 4, 2$

Step 2.7

Find the domain of $\frac{4 - x}{x - 2}$ .

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Step 2.7.1

Set the denominator in $\frac{4 - x}{x - 2}$ equal to $0$ to find where the expression is undefined.

$x - 2 = 0$

Step 2.7.2

Add $2$ to both sides of the equation.

$x = 2$

Step 2.7.3

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

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

Step 2.8

Use each root to create test intervals.

$x < 2$

$2 < x < 4$

$x > 4$

Step 2.9

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 2.9.1

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

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Step 2.9.1.1

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

$x = 0$

Step 2.9.1.2

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

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

Step 2.9.1.3

The left side $- 2$ is not greater than the right side $0$ , which means that the given statement is false.

False

Step 2.9.2

Test a value on the interval $2 < x < 4$ to see if it makes the inequality true.

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Step 2.9.2.1

Choose a value on the interval $2 < x < 4$ and see if this value makes the original inequality true.

$x = 3$

Step 2.9.2.2

Replace $x$ with $3$ in the original inequality.

$\frac{4 - (3)}{(3) - 2} > 0$

Step 2.9.2.3

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

True

Step 2.9.3

Test a value on the interval $x > 4$ to see if it makes the inequality true.

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Step 2.9.3.1

Choose a value on the interval $x > 4$ and see if this value makes the original inequality true.

$x = 6$

Step 2.9.3.2

Replace $x$ with $6$ in the original inequality.

$\frac{4 - (6)}{(6) - 2} > 0$

Step 2.9.3.3

The left side $- 0.5$ is not greater than the right side $0$ , which means that the given statement is false.

False

Step 2.9.4

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

$x < 2$ False

$2 < x < 4$ True

$x > 4$ False

$x < 2$ False

$2 < x < 4$ True

$x > 4$ False

Step 2.10

The solution consists of all of the true intervals.

$2 < x < 4$

Step 3

Set the radicand in $\sqrt{\ln (\frac{4 - x}{x - 2})}$ greater than or equal to $0$ to find where the expression is defined.

$\ln (\frac{4 - x}{x - 2}) \geq 0$

Step 4

Solve for $x$ .

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Step 4.1

Convert the inequality to an equality.

$\ln (\frac{4 - x}{x - 2}) = 0$

Step 4.2

Solve the equation.

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Step 4.2.1

Rewrite $\ln (\frac{4 - x}{x - 2}) = 0$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b$ $\neq$ $1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{0} = \frac{4 - x}{x - 2}$

Step 4.2.2

Cross multiply to remove the fraction.

$4 - x = e^{0} (x - 2)$

Step 4.2.3

Simplify $e^{0} (x - 2)$ .

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Step 4.2.3.1

Remove parentheses.

$4 - x = e^{0} (x - 2)$

Step 4.2.3.2

Anything raised to $0$ is $1$ .

$4 - x = 1 (x - 2)$

Step 4.2.3.3

Multiply $x - 2$ by $1$ .

$4 - x = x - 2$

Step 4.2.4

Move all terms containing $x$ to the left side of the equation.

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Step 4.2.4.1

Subtract $x$ from both sides of the equation.

$4 - x - x = - 2$

Step 4.2.4.2

Subtract $x$ from $- x$ .

$4 - 2 x = - 2$

Step 4.2.5

Factor $2$ out of $4 - 2 x$ .

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Step 4.2.5.1

Factor $2$ out of $4$ .

$2 (2) - 2 x = - 2$

Step 4.2.5.2

Factor $2$ out of $- 2 x$ .

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

Step 4.2.5.3

Factor $2$ out of $2 (2) + 2 (- x)$ .

$2 (2 - x) = - 2$

Step 4.2.6

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

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Step 4.2.6.1

Divide each term in $2 (2 - x) = - 2$ by $2$ .

$\frac{2 (2 - x)}{2} = \frac{- 2}{2}$

Step 4.2.6.2

Simplify the left side.

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Step 4.2.6.2.1

Cancel the common factor of $2$ .

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Step 4.2.6.2.1.1

Cancel the common factor.

$\frac{2 (2 - x)}{2} = \frac{- 2}{2}$

Step 4.2.6.2.1.2

Divide $2 - x$ by $1$ .

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

Step 4.2.6.3

Simplify the right side.

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Step 4.2.6.3.1

Divide $- 2$ by $2$ .

$2 - x = - 1$

Step 4.2.7

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

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Step 4.2.7.1

Subtract $2$ from both sides of the equation.

$- x = - 1 - 2$

Step 4.2.7.2

Subtract $2$ from $- 1$ .

$- x = - 3$

Step 4.2.8

Divide each term in $- x = - 3$ by $- 1$ and simplify.

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Step 4.2.8.1

Divide each term in $- x = - 3$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 3}{- 1}$

Step 4.2.8.2

Simplify the left side.

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Step 4.2.8.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 3}{- 1}$

Step 4.2.8.2.2

Divide $x$ by $1$ .

$x = \frac{- 3}{- 1}$

Step 4.2.8.3

Simplify the right side.

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Step 4.2.8.3.1

Divide $- 3$ by $- 1$ .

$x = 3$

Step 4.3

Find the domain of $\ln (\frac{4 - x}{x - 2})$ .

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Step 4.3.1

Set the argument in $\ln (\frac{4 - x}{x - 2})$ greater than $0$ to find where the expression is defined.

$\frac{4 - x}{x - 2} > 0$

Step 4.3.2

Solve for $x$ .

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Step 4.3.2.1

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

$4 - x = 0$

$x - 2 = 0$

Step 4.3.2.2

Subtract $4$ from both sides of the equation.

$- x = - 4$

Step 4.3.2.3

Divide each term in $- x = - 4$ by $- 1$ and simplify.

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Step 4.3.2.3.1

Divide each term in $- x = - 4$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 4}{- 1}$

Step 4.3.2.3.2

Simplify the left side.

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Step 4.3.2.3.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 4}{- 1}$

Step 4.3.2.3.2.2

Divide $x$ by $1$ .

$x = \frac{- 4}{- 1}$

Step 4.3.2.3.3

Simplify the right side.

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Step 4.3.2.3.3.1

Divide $- 4$ by $- 1$ .

$x = 4$

Step 4.3.2.4

Add $2$ to both sides of the equation.

$x = 2$

Step 4.3.2.5

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

$x = 4$

$x = 2$

Step 4.3.2.6

Consolidate the solutions.

$x = 4, 2$

Step 4.3.2.7

Find the domain of $\frac{4 - x}{x - 2}$ .

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Step 4.3.2.7.1

Set the denominator in $\frac{4 - x}{x - 2}$ equal to $0$ to find where the expression is undefined.

$x - 2 = 0$

Step 4.3.2.7.2

Add $2$ to both sides of the equation.

$x = 2$

Step 4.3.2.7.3

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

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

Step 4.3.2.8

Use each root to create test intervals.

$x < 2$

$2 < x < 4$

$x > 4$

Step 4.3.2.9

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 4.3.2.9.1

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

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Step 4.3.2.9.1.1

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

$x = 0$

Step 4.3.2.9.1.2

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

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

Step 4.3.2.9.1.3

The left side $- 2$ is not greater than the right side $0$ , which means that the given statement is false.

False

Step 4.3.2.9.2

Test a value on the interval $2 < x < 4$ to see if it makes the inequality true.

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Step 4.3.2.9.2.1

Choose a value on the interval $2 < x < 4$ and see if this value makes the original inequality true.

$x = 3$

Step 4.3.2.9.2.2

Replace $x$ with $3$ in the original inequality.

$\frac{4 - (3)}{(3) - 2} > 0$

Step 4.3.2.9.2.3

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

True

Step 4.3.2.9.3

Test a value on the interval $x > 4$ to see if it makes the inequality true.

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Step 4.3.2.9.3.1

Choose a value on the interval $x > 4$ and see if this value makes the original inequality true.

$x = 6$

Step 4.3.2.9.3.2

Replace $x$ with $6$ in the original inequality.

$\frac{4 - (6)}{(6) - 2} > 0$

Step 4.3.2.9.3.3

The left side $- 0.5$ is not greater than the right side $0$ , which means that the given statement is false.

False

Step 4.3.2.9.4

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

$x < 2$ False

$2 < x < 4$ True

$x > 4$ False

$x < 2$ False

$2 < x < 4$ True

$x > 4$ False

Step 4.3.2.10

The solution consists of all of the true intervals.

$2 < x < 4$

Step 4.3.3

Set the denominator in $\frac{4 - x}{x - 2}$ equal to $0$ to find where the expression is undefined.

$x - 2 = 0$

Step 4.3.4

Add $2$ to both sides of the equation.

$x = 2$

Step 4.3.5

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

$(2, 4)$

Step 4.4

Use each root to create test intervals.

$x < 2$

$2 < x < 3$

$3 < x < 4$

$x > 4$

Step 4.5

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 4.5.1

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

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Step 4.5.1.1

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

$x = 0$

Step 4.5.1.2

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

$\ln (\frac{4 - (0)}{(0) - 2}) \geq 0$

Step 4.5.1.3

Determine if the inequality is true.

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Step 4.5.1.3.1

The equation cannot be solved because it is undefined.

$Undefined$

Step 4.5.1.3.2

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

False

Step 4.5.2

Test a value on the interval $2 < x < 3$ to see if it makes the inequality true.

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Step 4.5.2.1

Choose a value on the interval $2 < x < 3$ and see if this value makes the original inequality true.

$x = 2.5$

Step 4.5.2.2

Replace $x$ with $2.5$ in the original inequality.

$\ln (\frac{4 - (2.5)}{(2.5) - 2}) \geq 0$

Step 4.5.2.3

The left side $1.09861228$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 4.5.3

Test a value on the interval $3 < x < 4$ to see if it makes the inequality true.

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Step 4.5.3.1

Choose a value on the interval $3 < x < 4$ and see if this value makes the original inequality true.

$x = 3.5$

Step 4.5.3.2

Replace $x$ with $3.5$ in the original inequality.

$\ln (\frac{4 - (3.5)}{(3.5) - 2}) \geq 0$

Step 4.5.3.3

The left side $- 1.09861228$ is less than the right side $0$ , which means that the given statement is false.

False

Step 4.5.4

Test a value on the interval $x > 4$ to see if it makes the inequality true.

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Step 4.5.4.1

Choose a value on the interval $x > 4$ and see if this value makes the original inequality true.

$x = 6$

Step 4.5.4.2

Replace $x$ with $6$ in the original inequality.

$\ln (\frac{4 - (6)}{(6) - 2}) \geq 0$

Step 4.5.4.3

Determine if the inequality is true.

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Step 4.5.4.3.1

The equation cannot be solved because it is undefined.

$Undefined$

Step 4.5.4.3.2

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

False

Step 4.5.5

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

$x < 2$ False

$2 < x < 3$ True

$3 < x < 4$ False

$x > 4$ False

$x < 2$ False

$2 < x < 3$ True

$3 < x < 4$ False

$x > 4$ False

Step 4.6

The solution consists of all of the true intervals.

$2 < x \leq 3$

Step 5

Set the denominator in $\frac{4 - x}{x - 2}$ equal to $0$ to find where the expression is undefined.

$x - 2 = 0$

Step 6

Add $2$ to both sides of the equation.

$x = 2$

Step 7

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

Interval Notation:

$(2, 3]$

Set-Builder Notation:

${x | 2 < x \leq 3}$

Step 8