Precalculus Examples

$\log (x - 2) + \log (9 - x) < 1$

Step 1

Convert the inequality to an equality.

$\log (x - 2) + \log (9 - x) = 1$

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, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\log ((x - 2) (9 - x)) = 1$

Step 2.1.2

Expand $(x - 2) (9 - x)$ using the FOIL Method.

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Step 2.1.2.1

Apply the distributive property.

$\log (x (9 - x) - 2 (9 - x)) = 1$

Step 2.1.2.2

Apply the distributive property.

$\log (x \cdot 9 + x (- x) - 2 (9 - x)) = 1$

Step 2.1.2.3

Apply the distributive property.

$\log (x \cdot 9 + x (- x) - 2 \cdot 9 - 2 (- x)) = 1$

Step 2.1.3

Simplify and combine like terms.

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Step 2.1.3.1

Simplify each term.

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Step 2.1.3.1.1

Move $9$ to the left of $x$ .

$\log (9 \cdot x + x (- x) - 2 \cdot 9 - 2 (- x)) = 1$

Step 2.1.3.1.2

Rewrite using the commutative property of multiplication.

$\log (9 x - x \cdot x - 2 \cdot 9 - 2 (- x)) = 1$

Step 2.1.3.1.3

Multiply $x$ by $x$ by adding the exponents.

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Step 2.1.3.1.3.1

Move $x$ .

$\log (9 x - (x \cdot x) - 2 \cdot 9 - 2 (- x)) = 1$

Step 2.1.3.1.3.2

Multiply $x$ by $x$ .

$\log (9 x - x^{2} - 2 \cdot 9 - 2 (- x)) = 1$

Step 2.1.3.1.4

Multiply $- 2$ by $9$ .

$\log (9 x - x^{2} - 18 - 2 (- x)) = 1$

Step 2.1.3.1.5

Multiply $- 1$ by $- 2$ .

$\log (9 x - x^{2} - 18 + 2 x) = 1$

Step 2.1.3.2

Add $9 x$ and $2 x$ .

$\log (11 x - x^{2} - 18) = 1$

Step 2.2

Rewrite $\log (11 x - x^{2} - 18) = 1$ 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 $b^{y} = x$ .

$10^{1} = 11 x - x^{2} - 18$

Step 2.3

Solve for $x$ .

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Step 2.3.1

Rewrite the equation as $11 x - x^{2} - 18 = 10^{1}$ .

$11 x - x^{2} - 18 = 10$

Step 2.3.2

Subtract $10$ from both sides of the equation.

$11 x - x^{2} - 18 - 10 = 0$

Step 2.3.3

Subtract $10$ from $- 18$ .

$11 x - x^{2} - 28 = 0$

Step 2.3.4

Factor the left side of the equation.

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Step 2.3.4.1

Factor $- 1$ out of $11 x - x^{2} - 28$ .

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Step 2.3.4.1.1

Reorder $11 x$ and $- x^{2}$ .

$- x^{2} + 11 x - 28 = 0$

Step 2.3.4.1.2

Factor $- 1$ out of $- x^{2}$ .

$- (x^{2}) + 11 x - 28 = 0$

Step 2.3.4.1.3

Factor $- 1$ out of $11 x$ .

$- (x^{2}) - (- 11 x) - 28 = 0$

Step 2.3.4.1.4

Rewrite $- 28$ as $- 1 (28)$ .

$- (x^{2}) - (- 11 x) - 1 \cdot 28 = 0$

Step 2.3.4.1.5

Factor $- 1$ out of $- (x^{2}) - (- 11 x)$ .

$- (x^{2} - 11 x) - 1 \cdot 28 = 0$

Step 2.3.4.1.6

Factor $- 1$ out of $- (x^{2} - 11 x) - 1 (28)$ .

$- (x^{2} - 11 x + 28) = 0$

Step 2.3.4.2

Factor.

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Step 2.3.4.2.1

Factor $x^{2} - 11 x + 28$ using the AC method.

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Step 2.3.4.2.1.1

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 $28$ and whose sum is $- 11$ .

$- 7, - 4$

Step 2.3.4.2.1.2

Write the factored form using these integers.

$- ((x - 7) (x - 4)) = 0$

Step 2.3.4.2.2

Remove unnecessary parentheses.

$- (x - 7) (x - 4) = 0$

Step 2.3.5

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

$x - 7 = 0$

$x - 4 = 0$

Step 2.3.6

Set $x - 7$ equal to $0$ and solve for $x$ .

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Step 2.3.6.1

Set $x - 7$ equal to $0$ .

$x - 7 = 0$

Step 2.3.6.2

Add $7$ to both sides of the equation.

$x = 7$

Step 2.3.7

Set $x - 4$ equal to $0$ and solve for $x$ .

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Step 2.3.7.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 2.3.7.2

Add $4$ to both sides of the equation.

$x = 4$

Step 2.3.8

The final solution is all the values that make $- (x - 7) (x - 4) = 0$ true.

$x = 7, 4$

Step 3

Find the domain of $\log (11 x - x^{2} - 18) - 1$ .

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Step 3.1

Set the argument in $\log (11 x - x^{2} - 18)$ greater than $0$ to find where the expression is defined.

$11 x - x^{2} - 18 > 0$

Step 3.2

Solve for $x$ .

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Step 3.2.1

Convert the inequality to an equation.

$11 x - x^{2} - 18 = 0$

Step 3.2.2

Factor the left side of the equation.

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Step 3.2.2.1

Factor $- 1$ out of $11 x - x^{2} - 18$ .

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Step 3.2.2.1.1

Reorder $11 x$ and $- x^{2}$ .

$- x^{2} + 11 x - 18 = 0$

Step 3.2.2.1.2

Factor $- 1$ out of $- x^{2}$ .

$- (x^{2}) + 11 x - 18 = 0$

Step 3.2.2.1.3

Factor $- 1$ out of $11 x$ .

$- (x^{2}) - (- 11 x) - 18 = 0$

Step 3.2.2.1.4

Rewrite $- 18$ as $- 1 (18)$ .

$- (x^{2}) - (- 11 x) - 1 \cdot 18 = 0$

Step 3.2.2.1.5

Factor $- 1$ out of $- (x^{2}) - (- 11 x)$ .

$- (x^{2} - 11 x) - 1 \cdot 18 = 0$

Step 3.2.2.1.6

Factor $- 1$ out of $- (x^{2} - 11 x) - 1 (18)$ .

$- (x^{2} - 11 x + 18) = 0$

Step 3.2.2.2

Factor.

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Step 3.2.2.2.1

Factor $x^{2} - 11 x + 18$ using the AC method.

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Step 3.2.2.2.1.1

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 $18$ and whose sum is $- 11$ .

$- 9, - 2$

Step 3.2.2.2.1.2

Write the factored form using these integers.

$- ((x - 9) (x - 2)) = 0$

Step 3.2.2.2.2

Remove unnecessary parentheses.

$- (x - 9) (x - 2) = 0$

Step 3.2.3

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

$x - 9 = 0$

$x - 2 = 0$

Step 3.2.4

Set $x - 9$ equal to $0$ and solve for $x$ .

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Step 3.2.4.1

Set $x - 9$ equal to $0$ .

$x - 9 = 0$

Step 3.2.4.2

Add $9$ to both sides of the equation.

$x = 9$

Step 3.2.5

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

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Step 3.2.5.1

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

$x - 2 = 0$

Step 3.2.5.2

Add $2$ to both sides of the equation.

$x = 2$

Step 3.2.6

The final solution is all the values that make $- (x - 9) (x - 2) = 0$ true.

$x = 9, 2$

Step 3.2.7

Use each root to create test intervals.

$x < 2$

$2 < x < 9$

$x > 9$

Step 3.2.8

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 3.2.8.1

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

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Step 3.2.8.1.1

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

$x = 0$

Step 3.2.8.1.2

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

$11 (0) - {(0)}^{2} - 18 > 0$

Step 3.2.8.1.3

The left side $- 18$ is not greater than the right side $0$ , which means that the given statement is false.

False

Step 3.2.8.2

Test a value on the interval $2 < x < 9$ to see if it makes the inequality true.

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Step 3.2.8.2.1

Choose a value on the interval $2 < x < 9$ and see if this value makes the original inequality true.

$x = 6$

Step 3.2.8.2.2

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

$11 (6) - {(6)}^{2} - 18 > 0$

Step 3.2.8.2.3

The left side $12$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 3.2.8.3

Test a value on the interval $x > 9$ to see if it makes the inequality true.

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Step 3.2.8.3.1

Choose a value on the interval $x > 9$ and see if this value makes the original inequality true.

$x = 12$

Step 3.2.8.3.2

Replace $x$ with $12$ in the original inequality.

$11 (12) - {(12)}^{2} - 18 > 0$

Step 3.2.8.3.3

The left side $- 30$ is not greater than the right side $0$ , which means that the given statement is false.

False

Step 3.2.8.4

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

$x < 2$ False

$2 < x < 9$ True

$x > 9$ False

$x < 2$ False

$2 < x < 9$ True

$x > 9$ False

Step 3.2.9

The solution consists of all of the true intervals.

$2 < x < 9$

Step 3.3

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

$(2, 9)$

Step 4

Use each root to create test intervals.

$x < 2$

$2 < x < 4$

$4 < x < 7$

$7 < x < 9$

$x > 9$

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.

$x = 0$

Step 5.1.2

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

$\log ((0) - 2) + \log (9 - (0)) < 1$

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 $2 < x < 4$ to see if it makes the inequality true.

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Step 5.2.1

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

$x = 3$

Step 5.2.2

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

$\log ((3) - 2) + \log (9 - (3)) < 1$

Step 5.2.3

The left side $0.77815125$ is less than the right side $1$ , which means that the given statement is always true.

True

Step 5.3

Test a value on the interval $4 < x < 7$ to see if it makes the inequality true.

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Step 5.3.1

Choose a value on the interval $4 < x < 7$ and see if this value makes the original inequality true.

$x = 6$

Step 5.3.2

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

$\log ((6) - 2) + \log (9 - (6)) < 1$

Step 5.3.3

The left side $1.07918124$ is not less than the right side $1$ , which means that the given statement is false.

False

Step 5.4

Test a value on the interval $7 < x < 9$ to see if it makes the inequality true.

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Step 5.4.1

Choose a value on the interval $7 < x < 9$ and see if this value makes the original inequality true.

$x = 8$

Step 5.4.2

Replace $x$ with $8$ in the original inequality.

$\log ((8) - 2) + \log (9 - (8)) < 1$

Step 5.4.3

The left side $0.77815125$ is less than the right side $1$ , which means that the given statement is always true.

True

Step 5.5

Test a value on the interval $x > 9$ to see if it makes the inequality true.

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Step 5.5.1

Choose a value on the interval $x > 9$ and see if this value makes the original inequality true.

$x = 12$

Step 5.5.2

Replace $x$ with $12$ in the original inequality.

$\log ((12) - 2) + \log (9 - (12)) < 1$

Step 5.5.3

Determine if the inequality is true.

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Step 5.5.3.1

The equation cannot be solved because it is undefined.

$Undefined$

Step 5.5.3.2

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

False

Step 5.6

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

$x < 2$ False

$2 < x < 4$ True

$4 < x < 7$ False

$7 < x < 9$ True

$x > 9$ False

$x < 2$ False

$2 < x < 4$ True

$4 < x < 7$ False

$7 < x < 9$ True

$x > 9$ False

Step 6

The solution consists of all of the true intervals.

$2 < x < 4$ or $7 < x < 9$

Step 7

The result can be shown in multiple forms.

Inequality Form:

$2 < x < 4 or 7 < x < 9$

Interval Notation:

$(2, 4) \cup (7, 9)$

Step 8