Algebra Examples

$\log (3 x - 1) > \log (4 - x)$

Step 1

Convert the inequality to an equality.

$\log (3 x - 1) = \log (4 - x)$

Step 2

Solve the equation.

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Step 2.1

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

$3 x - 1 = 4 - x$

Step 2.2

Solve for $x$ .

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Step 2.2.1

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

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Step 2.2.1.1

Add $x$ to both sides of the equation.

$3 x - 1 + x = 4$

Step 2.2.1.2

Add $3 x$ and $x$ .

$4 x - 1 = 4$

Step 2.2.2

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

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Step 2.2.2.1

Add $1$ to both sides of the equation.

$4 x = 4 + 1$

Step 2.2.2.2

Add $4$ and $1$ .

$4 x = 5$

Step 2.2.3

Divide each term in $4 x = 5$ by $4$ and simplify.

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Step 2.2.3.1

Divide each term in $4 x = 5$ by $4$ .

$\frac{4 x}{4} = \frac{5}{4}$

Step 2.2.3.2

Simplify the left side.

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Step 2.2.3.2.1

Cancel the common factor of $4$ .

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Step 2.2.3.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{5}{4}$

Step 2.2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{5}{4}$

Step 3

Find the domain of $\log (\frac{3 x - 1}{4 - x})$ .

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Step 3.1

Set the argument in $\log (\frac{3 x - 1}{4 - x})$ greater than $0$ to find where the expression is defined.

$\frac{3 x - 1}{4 - x} > 0$

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.

$3 x - 1 = 0$

$4 - x = 0$

Step 3.2.2

Add $1$ to both sides of the equation.

$3 x = 1$

Step 3.2.3

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

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Step 3.2.3.1

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

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

Step 3.2.3.2

Simplify the left side.

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Step 3.2.3.2.1

Cancel the common factor of $3$ .

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Step 3.2.3.2.1.1

Cancel the common factor.

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

Step 3.2.3.2.1.2

Divide $x$ by $1$ .

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

Step 3.2.4

Subtract $4$ from both sides of the equation.

$- x = - 4$

Step 3.2.5

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

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Step 3.2.5.1

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

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

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.

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

Step 3.2.5.2.2

Divide $x$ by $1$ .

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

Step 3.2.5.3

Simplify the right side.

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Step 3.2.5.3.1

Divide $- 4$ by $- 1$ .

$x = 4$

Step 3.2.6

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

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

$x = 4$

Step 3.2.7

Consolidate the solutions.

$x = \frac{1}{3}, 4$

Step 3.2.8

Find the domain of $\frac{3 x - 1}{4 - x}$ .

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Step 3.2.8.1

Set the denominator in $\frac{3 x - 1}{4 - x}$ equal to $0$ to find where the expression is undefined.

$4 - x = 0$

Step 3.2.8.2

Solve for $x$ .

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Step 3.2.8.2.1

Subtract $4$ from both sides of the equation.

$- x = - 4$

Step 3.2.8.2.2

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

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Step 3.2.8.2.2.1

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

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

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.

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

Step 3.2.8.2.2.2.2

Divide $x$ by $1$ .

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

Step 3.2.8.2.2.3

Simplify the right side.

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Step 3.2.8.2.2.3.1

Divide $- 4$ by $- 1$ .

$x = 4$

Step 3.2.8.3

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

$(- \infty, 4) \cup (4, \infty)$

Step 3.2.9

Use each root to create test intervals.

$x < \frac{1}{3}$

$\frac{1}{3} < x < 4$

$x > 4$

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 $x < \frac{1}{3}$ to see if it makes the inequality true.

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Step 3.2.10.1.1

Choose a value on the interval $x < \frac{1}{3}$ and see if this value makes the original inequality true.

$x = 0$

Step 3.2.10.1.2

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

$\frac{3 (0) - 1}{4 - (0)} > 0$

Step 3.2.10.1.3

The left side $- 0.25$ is not greater than the right side $0$ , which means that the given statement is false.

False

Step 3.2.10.2

Test a value on the interval $\frac{1}{3} < x < 4$ to see if it makes the inequality true.

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Step 3.2.10.2.1

Choose a value on the interval $\frac{1}{3} < x < 4$ and see if this value makes the original inequality true.

$x = 2$

Step 3.2.10.2.2

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

$\frac{3 (2) - 1}{4 - (2)} > 0$

Step 3.2.10.2.3

The left side $2.5$ 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 $x > 4$ to see if it makes the inequality true.

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Step 3.2.10.3.1

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

$x = 6$

Step 3.2.10.3.2

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

$\frac{3 (6) - 1}{4 - (6)} > 0$

Step 3.2.10.3.3

The left side $- 8.5$ 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.

$x < \frac{1}{3}$ False

$\frac{1}{3} < x < 4$ True

$x > 4$ False

$x < \frac{1}{3}$ False

$\frac{1}{3} < x < 4$ True

$x > 4$ False

Step 3.2.11

The solution consists of all of the true intervals.

$\frac{1}{3} < x < 4$

Step 3.3

Set the denominator in $\frac{3 x - 1}{4 - x}$ equal to $0$ to find where the expression is undefined.

$4 - x = 0$

Step 3.4

Solve for $x$ .

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Step 3.4.1

Subtract $4$ from both sides of the equation.

$- x = - 4$

Step 3.4.2

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

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Step 3.4.2.1

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

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

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.

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

Step 3.4.2.2.2

Divide $x$ by $1$ .

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

Step 3.4.2.3

Simplify the right side.

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Step 3.4.2.3.1

Divide $- 4$ by $- 1$ .

$x = 4$

Step 3.5

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

$(\frac{1}{3}, 4)$

Step 4

Use each root to create test intervals.

$x < \frac{1}{3}$

$\frac{1}{3} < x < \frac{5}{4}$

$\frac{5}{4} < x < 4$

$x > 4$

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 $x < \frac{1}{3}$ to see if it makes the inequality true.

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Step 5.1.1

Choose a value on the interval $x < \frac{1}{3}$ and see if this value makes the original inequality true.

$x = 0$

Step 5.1.2

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

$\log (3 (0) - 1) > \log (4 - (0))$

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 $\frac{1}{3} < x < \frac{5}{4}$ to see if it makes the inequality true.

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Step 5.2.1

Choose a value on the interval $\frac{1}{3} < x < \frac{5}{4}$ and see if this value makes the original inequality true.

$x = 1$

Step 5.2.2

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

$\log (3 (1) - 1) > \log (4 - (1))$

Step 5.2.3

The left side $0.30102999$ is not greater than the right side $0.47712125$ , which means that the given statement is false.

False

Step 5.3

Test a value on the interval $\frac{5}{4} < x < 4$ to see if it makes the inequality true.

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Step 5.3.1

Choose a value on the interval $\frac{5}{4} < x < 4$ and see if this value makes the original inequality true.

$x = 3$

Step 5.3.2

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

$\log (3 (3) - 1) > \log (4 - (3))$

Step 5.3.3

The left side $0.90308998$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 5.4

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

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Step 5.4.1

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

$x = 6$

Step 5.4.2

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

$\log (3 (6) - 1) > \log (4 - (6))$

Step 5.4.3

Determine if the inequality is true.

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Step 5.4.3.1

The equation cannot be solved because it is undefined.

$Undefined$

Step 5.4.3.2

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

False

Step 5.5

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

$x < \frac{1}{3}$ False

$\frac{1}{3} < x < \frac{5}{4}$ False

$\frac{5}{4} < x < 4$ True

$x > 4$ False

$x < \frac{1}{3}$ False

$\frac{1}{3} < x < \frac{5}{4}$ False

$\frac{5}{4} < x < 4$ True

$x > 4$ False

Step 6

The solution consists of all of the true intervals.

$\frac{5}{4} < x < 4$

Step 7

The result can be shown in multiple forms.

Inequality Form:

$\frac{5}{4} < x < 4$

Interval Notation:

$(\frac{5}{4}, 4)$

Step 8