Algebra Examples

$\log_{9} (4 x - 5) < \log_{9} (- 2 x + 1)$

Step 1

Convert the inequality to an equality.

$\log_{9} (4 x - 5) = \log_{9} (- 2 x + 1)$

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.

$4 x - 5 = - 2 x + 1$

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 $2 x$ to both sides of the equation.

$4 x - 5 + 2 x = 1$

Step 2.2.1.2

Add $4 x$ and $2 x$ .

$6 x - 5 = 1$

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 $5$ to both sides of the equation.

$6 x = 1 + 5$

Step 2.2.2.2

Add $1$ and $5$ .

$6 x = 6$

Step 2.2.3

Divide each term in $6 x = 6$ by $6$ and simplify.

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Step 2.2.3.1

Divide each term in $6 x = 6$ by $6$ .

$\frac{6 x}{6} = \frac{6}{6}$

Step 2.2.3.2

Simplify the left side.

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Step 2.2.3.2.1

Cancel the common factor of $6$ .

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Step 2.2.3.2.1.1

Cancel the common factor.

$\frac{6 x}{6} = \frac{6}{6}$

Step 2.2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{6}{6}$

Step 2.2.3.3

Simplify the right side.

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Step 2.2.3.3.1

Divide $6$ by $6$ .

$x = 1$

Step 3

Find the domain of $\log_{9} (\frac{4 x - 5}{- 2 x + 1})$ .

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Step 3.1

Set the argument in $\log_{9} (\frac{4 x - 5}{- 2 x + 1})$ greater than $0$ to find where the expression is defined.

$\frac{4 x - 5}{- 2 x + 1} > 0$

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.

$4 x - 5 = 0$

$- 2 x + 1 = 0$

Step 3.2.2

Add $5$ to both sides of the equation.

$4 x = 5$

Step 3.2.3

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

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Step 3.2.3.1

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

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

Step 3.2.3.2

Simplify the left side.

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Step 3.2.3.2.1

Cancel the common factor of $4$ .

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Step 3.2.3.2.1.1

Cancel the common factor.

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

Step 3.2.3.2.1.2

Divide $x$ by $1$ .

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

Step 3.2.4

Subtract $1$ from both sides of the equation.

$- 2 x = - 1$

Step 3.2.5

Divide each term in $- 2 x = - 1$ by $- 2$ and simplify.

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Step 3.2.5.1

Divide each term in $- 2 x = - 1$ by $- 2$ .

$\frac{- 2 x}{- 2} = \frac{- 1}{- 2}$

Step 3.2.5.2

Simplify the left side.

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Step 3.2.5.2.1

Cancel the common factor of $- 2$ .

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Step 3.2.5.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} = \frac{- 1}{- 2}$

Step 3.2.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{- 2}$

Step 3.2.5.3

Simplify the right side.

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Step 3.2.5.3.1

Dividing two negative values results in a positive value.

$x = \frac{1}{2}$

Step 3.2.6

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

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

$x = \frac{1}{2}$

Step 3.2.7

Consolidate the solutions.

$x = \frac{5}{4}, \frac{1}{2}$

Step 3.2.8

Find the domain of $\frac{4 x - 5}{- 2 x + 1}$ .

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Step 3.2.8.1

Set the denominator in $\frac{4 x - 5}{- 2 x + 1}$ equal to $0$ to find where the expression is undefined.

$- 2 x + 1 = 0$

Step 3.2.8.2

Solve for $x$ .

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Step 3.2.8.2.1

Subtract $1$ from both sides of the equation.

$- 2 x = - 1$

Step 3.2.8.2.2

Divide each term in $- 2 x = - 1$ by $- 2$ and simplify.

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Step 3.2.8.2.2.1

Divide each term in $- 2 x = - 1$ by $- 2$ .

$\frac{- 2 x}{- 2} = \frac{- 1}{- 2}$

Step 3.2.8.2.2.2

Simplify the left side.

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Step 3.2.8.2.2.2.1

Cancel the common factor of $- 2$ .

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Step 3.2.8.2.2.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} = \frac{- 1}{- 2}$

Step 3.2.8.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{- 2}$

Step 3.2.8.2.2.3

Simplify the right side.

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Step 3.2.8.2.2.3.1

Dividing two negative values results in a positive value.

$x = \frac{1}{2}$

Step 3.2.8.3

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

$(- \infty, \frac{1}{2}) \cup (\frac{1}{2}, \infty)$

Step 3.2.9

Use each root to create test intervals.

$x < \frac{1}{2}$

$\frac{1}{2} < x < \frac{5}{4}$

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

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 $x < \frac{1}{2}$ to see if it makes the inequality true.

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Step 3.2.10.1.1

Choose a value on the interval $x < \frac{1}{2}$ 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.

$\frac{4 (0) - 5}{- 2 \cdot 0 + 1} > 0$

Step 3.2.10.1.3

The left side $- 5$ 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 $\frac{1}{2} < x < \frac{5}{4}$ to see if it makes the inequality true.

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Step 3.2.10.2.1

Choose a value on the interval $\frac{1}{2} < x < \frac{5}{4}$ and see if this value makes the original inequality true.

$x = 1$

Step 3.2.10.2.2

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

$\frac{4 (1) - 5}{- 2 \cdot 1 + 1} > 0$

Step 3.2.10.2.3

The left side $1$ 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 $x > \frac{5}{4}$ to see if it makes the inequality true.

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Step 3.2.10.3.1

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

$x = 4$

Step 3.2.10.3.2

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

$\frac{4 (4) - 5}{- 2 \cdot 4 + 1} > 0$

Step 3.2.10.3.3

The left side $- 1.\overline{571428}$ 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.

$x < \frac{1}{2}$ False

$\frac{1}{2} < x < \frac{5}{4}$ True

$x > \frac{5}{4}$ False

$x < \frac{1}{2}$ False

$\frac{1}{2} < x < \frac{5}{4}$ True

$x > \frac{5}{4}$ False

Step 3.2.11

The solution consists of all of the true intervals.

$\frac{1}{2} < x < \frac{5}{4}$

Step 3.3

Set the denominator in $\frac{4 x - 5}{- 2 x + 1}$ equal to $0$ to find where the expression is undefined.

$- 2 x + 1 = 0$

Step 3.4

Solve for $x$ .

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Step 3.4.1

Subtract $1$ from both sides of the equation.

$- 2 x = - 1$

Step 3.4.2

Divide each term in $- 2 x = - 1$ by $- 2$ and simplify.

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Step 3.4.2.1

Divide each term in $- 2 x = - 1$ by $- 2$ .

$\frac{- 2 x}{- 2} = \frac{- 1}{- 2}$

Step 3.4.2.2

Simplify the left side.

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Step 3.4.2.2.1

Cancel the common factor of $- 2$ .

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Step 3.4.2.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} = \frac{- 1}{- 2}$

Step 3.4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{- 2}$

Step 3.4.2.3

Simplify the right side.

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Step 3.4.2.3.1

Dividing two negative values results in a positive value.

$x = \frac{1}{2}$

Step 3.5

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

$(\frac{1}{2}, \frac{5}{4})$

Step 4

Use each root to create test intervals.

$x < \frac{1}{2}$

$\frac{1}{2} < x < 1$

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

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

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 $x < \frac{1}{2}$ to see if it makes the inequality true.

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Step 5.1.1

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

$x = 0$

Step 5.1.2

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

$\log_{9} (4 (0) - 5) < \log_{9} (- 2 \cdot 0 + 1)$

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 $\frac{1}{2} < x < 1$ to see if it makes the inequality true.

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Step 5.2.1

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

$x = 0.75$

Step 5.2.2

Replace $x$ with $0.75$ in the original inequality.

$\log_{9} (4 (0.75) - 5) < \log_{9} (- 2 \cdot 0.75 + 1)$

Step 5.2.3

Determine if the inequality is true.

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Step 5.2.3.1

The equation cannot be solved because it is undefined.

$Undefined$

Step 5.2.3.2

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

False

Step 5.3

Test a value on the interval $1 < x < \frac{5}{4}$ to see if it makes the inequality true.

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Step 5.3.1

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

$x = 1.12$

Step 5.3.2

Replace $x$ with $1.12$ in the original inequality.

$\log_{9} (4 (1.12) - 5) < \log_{9} (- 2 \cdot 1.12 + 1)$

Step 5.3.3

Determine if the inequality is true.

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Step 5.3.3.1

The equation cannot be solved because it is undefined.

$Undefined$

Step 5.3.3.2

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

False

Step 5.4

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

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Step 5.4.1

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

$x = 4$

Step 5.4.2

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

$\log_{9} (4 (4) - 5) < \log_{9} (- 2 \cdot 4 + 1)$

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.

$x < \frac{1}{2}$ False

$\frac{1}{2} < x < 1$ False

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

$x > \frac{5}{4}$ False

$x < \frac{1}{2}$ False

$\frac{1}{2} < x < 1$ False

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

$x > \frac{5}{4}$ False

Step 6

Since there are no numbers that fall within the interval, this inequality has no solution.

No solution