Finite Math Examples

$\log (x - 2) + \log (x + 2) > 2 \log (x - 1)$

Step 1

Convert the inequality to an equality.

$\log (x - 2) + \log (x + 2) = 2 \log (x - 1)$

Step 2

Solve the equation.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\log ((x - 2) (x + 2)) = 2 \log (x - 1)$

Step 2.1.2

Expand $(x - 2) (x + 2)$ using the FOIL Method.

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Step 2.1.2.1

Apply the distributive property.

$\log (x (x + 2) - 2 (x + 2)) = 2 \log (x - 1)$

Step 2.1.2.2

Apply the distributive property.

$\log (x \cdot x + x \cdot 2 - 2 (x + 2)) = 2 \log (x - 1)$

Step 2.1.2.3

Apply the distributive property.

$\log (x \cdot x + x \cdot 2 - 2 x - 2 \cdot 2) = 2 \log (x - 1)$

Step 2.1.3

Simplify terms.

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Step 2.1.3.1

Combine the opposite terms in $x \cdot x + x \cdot 2 - 2 x - 2 \cdot 2$ .

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Step 2.1.3.1.1

Reorder the factors in the terms $x \cdot 2$ and $- 2 x$ .

$\log (x \cdot x + 2 x - 2 x - 2 \cdot 2) = 2 \log (x - 1)$

Step 2.1.3.1.2

Subtract $2 x$ from $2 x$ .

$\log (x \cdot x + 0 - 2 \cdot 2) = 2 \log (x - 1)$

Step 2.1.3.1.3

Add $x \cdot x$ and $0$ .

$\log (x \cdot x - 2 \cdot 2) = 2 \log (x - 1)$

Step 2.1.3.2

Simplify each term.

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Step 2.1.3.2.1

Multiply $x$ by $x$ .

$\log (x^{2} - 2 \cdot 2) = 2 \log (x - 1)$

Step 2.1.3.2.2

Multiply $- 2$ by $2$ .

$\log (x^{2} - 4) = 2 \log (x - 1)$

Step 2.2

Simplify $2 \log (x - 1)$ by moving $2$ inside the logarithm.

$\log (x^{2} - 4) = \log ({(x - 1)}^{2})$

Step 2.3

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

$x^{2} - 4 = {(x - 1)}^{2}$

Step 2.4

Solve for $x$ .

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Step 2.4.1

Simplify ${(x - 1)}^{2}$ .

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Step 2.4.1.1

Rewrite.

$x^{2} - 4 = 0 + 0 + {(x - 1)}^{2}$

Step 2.4.1.2

Rewrite ${(x - 1)}^{2}$ as $(x - 1) (x - 1)$ .

$x^{2} - 4 = (x - 1) (x - 1)$

Step 2.4.1.3

Expand $(x - 1) (x - 1)$ using the FOIL Method.

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Step 2.4.1.3.1

Apply the distributive property.

$x^{2} - 4 = x (x - 1) - 1 (x - 1)$

Step 2.4.1.3.2

Apply the distributive property.

$x^{2} - 4 = x \cdot x + x \cdot - 1 - 1 (x - 1)$

Step 2.4.1.3.3

Apply the distributive property.

$x^{2} - 4 = x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1$

Step 2.4.1.4

Simplify and combine like terms.

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Step 2.4.1.4.1

Simplify each term.

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Step 2.4.1.4.1.1

Multiply $x$ by $x$ .

$x^{2} - 4 = x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1$

Step 2.4.1.4.1.2

Move $- 1$ to the left of $x$ .

$x^{2} - 4 = x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1$

Step 2.4.1.4.1.3

Rewrite $- 1 x$ as $- x$ .

$x^{2} - 4 = x^{2} - x - 1 x - 1 \cdot - 1$

Step 2.4.1.4.1.4

Rewrite $- 1 x$ as $- x$ .

$x^{2} - 4 = x^{2} - x - x - 1 \cdot - 1$

Step 2.4.1.4.1.5

Multiply $- 1$ by $- 1$ .

$x^{2} - 4 = x^{2} - x - x + 1$

Step 2.4.1.4.2

Subtract $x$ from $- x$ .

$x^{2} - 4 = x^{2} - 2 x + 1$

Step 2.4.2

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$x^{2} - 2 x + 1 = x^{2} - 4$

Step 2.4.3

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

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Step 2.4.3.1

Subtract $x^{2}$ from both sides of the equation.

$x^{2} - 2 x + 1 - x^{2} = - 4$

Step 2.4.3.2

Combine the opposite terms in $x^{2} - 2 x + 1 - x^{2}$ .

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Step 2.4.3.2.1

Subtract $x^{2}$ from $x^{2}$ .

$- 2 x + 1 + 0 = - 4$

Step 2.4.3.2.2

Add $- 2 x + 1$ and $0$ .

$- 2 x + 1 = - 4$

Step 2.4.4

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

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Step 2.4.4.1

Subtract $1$ from both sides of the equation.

$- 2 x = - 4 - 1$

Step 2.4.4.2

Subtract $1$ from $- 4$ .

$- 2 x = - 5$

Step 2.4.5

Divide each term in $- 2 x = - 5$ by $- 2$ and simplify.

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Step 2.4.5.1

Divide each term in $- 2 x = - 5$ by $- 2$ .

$\frac{- 2 x}{- 2} = \frac{- 5}{- 2}$

Step 2.4.5.2

Simplify the left side.

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Step 2.4.5.2.1

Cancel the common factor of $- 2$ .

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Step 2.4.5.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} = \frac{- 5}{- 2}$

Step 2.4.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 5}{- 2}$

Step 2.4.5.3

Simplify the right side.

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Step 2.4.5.3.1

Dividing two negative values results in a positive value.

$x = \frac{5}{2}$

Step 3

Find the domain of $\log (\frac{(x + 2) (x - 2)}{{(x - 1)}^{2}})$ .

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Step 3.1

Set the argument in $\log (\frac{(x + 2) (x - 2)}{{(x - 1)}^{2}})$ greater than $0$ to find where the expression is defined.

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

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.

$x + 2 = 0$

$x - 2 = 0$

${(x - 1)}^{2} = 0$

Step 3.2.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 3.2.3

Add $2$ to both sides of the equation.

$x = 2$

Step 3.2.4

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

$x - 1 = 0$

Step 3.2.5

Add $1$ to both sides of the equation.

$x = 1$

Step 3.2.6

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

$x = - 2$

$x = 2$

$x = 1$

Step 3.2.7

Consolidate the solutions.

$x = - 2, 2, 1$

Step 3.2.8

Find the domain of $\frac{(x + 2) (x - 2)}{{(x - 1)}^{2}}$ .

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Step 3.2.8.1

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

${(x - 1)}^{2} = 0$

Step 3.2.8.2

Solve for $x$ .

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Step 3.2.8.2.1

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

$x - 1 = 0$

Step 3.2.8.2.2

Add $1$ to both sides of the equation.

$x = 1$

Step 3.2.8.3

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

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

Step 3.2.9

Use each root to create test intervals.

$x < - 2$

$- 2 < x < 1$

$1 < x < 2$

$x > 2$

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 $x < - 2$ to see if it makes the inequality true.

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Step 3.2.10.1.1

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

$x = - 4$

Step 3.2.10.1.2

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

$\frac{((- 4) + 2) ((- 4) - 2)}{{((- 4) - 1)}^{2}} > 0$

Step 3.2.10.1.3

The left side $0.48$ 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 $- 2 < x < 1$ to see if it makes the inequality true.

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Step 3.2.10.2.1

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

$x = 0$

Step 3.2.10.2.2

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

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

Step 3.2.10.2.3

The left side $- 4$ is not greater than the right side $0$ , which means that the given statement is false.

False

Step 3.2.10.3

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

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Step 3.2.10.3.1

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

$x = 1.5$

Step 3.2.10.3.2

Replace $x$ with $1.5$ in the original inequality.

$\frac{((1.5) + 2) ((1.5) - 2)}{{((1.5) - 1)}^{2}} > 0$

Step 3.2.10.3.3

The left side $- 7$ is not greater than the right side $0$ , which means that the given statement is false.

False

Step 3.2.10.4

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

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Step 3.2.10.4.1

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

$x = 4$

Step 3.2.10.4.2

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

$\frac{((4) + 2) ((4) - 2)}{{((4) - 1)}^{2}} > 0$

Step 3.2.10.4.3

The left side $1.\overline{3}$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 3.2.10.5

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

$x < - 2$ True

$- 2 < x < 1$ False

$1 < x < 2$ False

$x > 2$ True

$x < - 2$ True

$- 2 < x < 1$ False

$1 < x < 2$ False

$x > 2$ True

Step 3.2.11

The solution consists of all of the true intervals.

$x < - 2$ or $x > 2$

Step 3.3

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

${(x - 1)}^{2} = 0$

Step 3.4

Solve for $x$ .

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Step 3.4.1

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

$x - 1 = 0$

Step 3.4.2

Add $1$ to both sides of the equation.

$x = 1$

Step 3.5

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

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

Step 4

The solution consists of all of the true intervals.

$x > \frac{5}{2}$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$x > \frac{5}{2}$

Interval Notation:

$(\frac{5}{2}, \infty)$

Step 6