Precalculus Examples

$\log_{3} (3 x - 6) < 2$

Step 1

Convert the inequality to an equality.

$\log_{3} (3 x - 6) = 2$

Step 2

Solve the equation.

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Step 2.1

Rewrite $\log_{3} (3 x - 6) = 2$ 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 $b^{y} = x$ .

$3^{2} = 3 x - 6$

Step 2.2

Solve for $x$ .

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Step 2.2.1

Rewrite the equation as $3 x - 6 = 3^{2}$ .

$3 x - 6 = 3^{2}$

Step 2.2.2

Raise $3$ to the power of $2$ .

$3 x - 6 = 9$

Step 2.2.3

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

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Step 2.2.3.1

Add $6$ to both sides of the equation.

$3 x = 9 + 6$

Step 2.2.3.2

Add $9$ and $6$ .

$3 x = 15$

Step 2.2.4

Divide each term in $3 x = 15$ by $3$ and simplify.

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Step 2.2.4.1

Divide each term in $3 x = 15$ by $3$ .

$\frac{3 x}{3} = \frac{15}{3}$

Step 2.2.4.2

Simplify the left side.

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Step 2.2.4.2.1

Cancel the common factor of $3$ .

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Step 2.2.4.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{15}{3}$

Step 2.2.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{15}{3}$

Step 2.2.4.3

Simplify the right side.

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Step 2.2.4.3.1

Divide $15$ by $3$ .

$x = 5$

Step 3

Find the domain of $\log_{3} (3 x - 6) - 2$ .

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Step 3.1

Set the argument in $\log_{3} (3 x - 6)$ greater than $0$ to find where the expression is defined.

$3 x - 6 > 0$

Step 3.2

Solve for $x$ .

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Step 3.2.1

Add $6$ to both sides of the inequality.

$3 x > 6$

Step 3.2.2

Divide each term in $3 x > 6$ by $3$ and simplify.

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Step 3.2.2.1

Divide each term in $3 x > 6$ by $3$ .

$\frac{3 x}{3} > \frac{6}{3}$

Step 3.2.2.2

Simplify the left side.

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Step 3.2.2.2.1

Cancel the common factor of $3$ .

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Step 3.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} > \frac{6}{3}$

Step 3.2.2.2.1.2

Divide $x$ by $1$ .

$x > \frac{6}{3}$

Step 3.2.2.3

Simplify the right side.

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Step 3.2.2.3.1

Divide $6$ by $3$ .

$x > 2$

Step 3.3

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

$(2, \infty)$

Step 4

Use each root to create test intervals.

$x < 2$

$2 < x < 5$

$x > 5$

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.

$\log_{3} (3 (0) - 6) < 2$

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 $2 < x < 5$ to see if it makes the inequality true.

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Step 5.2.1

Choose a value on the interval $2 < x < 5$ and see if this value makes the original inequality true.

$x = 4$

Step 5.2.2

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

$\log_{3} (3 (4) - 6) < 2$

Step 5.2.3

The left side $1.63092975$ is less than the right side $2$ , which means that the given statement is always true.

True

Step 5.3

Test a value on the interval $x > 5$ to see if it makes the inequality true.

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Step 5.3.1

Choose a value on the interval $x > 5$ and see if this value makes the original inequality true.

$x = 8$

Step 5.3.2

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

$\log_{3} (3 (8) - 6) < 2$

Step 5.3.3

The left side $2.63092975$ is not less than the right side $2$ , 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

$2 < x < 5$ True

$x > 5$ False

$x < 2$ False

$2 < x < 5$ True

$x > 5$ False

Step 6

The solution consists of all of the true intervals.

$2 < x < 5$

Step 7

The result can be shown in multiple forms.

Inequality Form:

$2 < x < 5$

Interval Notation:

$(2, 5)$

Step 8