Trigonometry Examples

$\log_{2} (x + 1) < \log_{2} (6 - 2 x)$

Step 1

Convert the inequality to an equality.

$\log_{2} (x + 1) = \log_{2} (6 - 2 x)$

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.

$x + 1 = 6 - 2 x$

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 $2 x$ to both sides of the equation.

$x + 1 + 2 x = 6$

Step 2.2.1.2

Add $x$ and $2 x$ .

$3 x + 1 = 6$

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

Subtract $1$ from both sides of the equation.

$3 x = 6 - 1$

Step 2.2.2.2

Subtract $1$ from $6$ .

$3 x = 5$

Step 2.2.3

Divide each term in $3 x = 5$ by $3$ and simplify.

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Step 2.2.3.1

Divide each term in $3 x = 5$ by $3$ .

$\frac{3 x}{3} = \frac{5}{3}$

Step 2.2.3.2

Simplify the left side.

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Step 2.2.3.2.1

Cancel the common factor of $3$ .

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Step 2.2.3.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{5}{3}$

Step 2.2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{5}{3}$

Step 3

Find the domain of $\log_{2} (\frac{x + 1}{2 (3 - x)})$ .

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Step 3.1

Set the argument in $\log_{2} (\frac{x + 1}{2 (3 - x)})$ greater than $0$ to find where the expression is defined.

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

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.

$x + 1 = 0$

$3 - x = 0$

Step 3.2.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 3.2.3

Subtract $3$ from both sides of the equation.

$- x = - 3$

Step 3.2.4

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

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Step 3.2.4.1

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

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

Step 3.2.4.2

Simplify the left side.

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Step 3.2.4.2.1

Dividing two negative values results in a positive value.

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

Step 3.2.4.2.2

Divide $x$ by $1$ .

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

Step 3.2.4.3

Simplify the right side.

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Step 3.2.4.3.1

Divide $- 3$ by $- 1$ .

$x = 3$

Step 3.2.5

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

$x = - 1$

$x = 3$

Step 3.2.6

Consolidate the solutions.

$x = - 1, 3$

Step 3.2.7

Find the domain of $\frac{x + 1}{2 (3 - x)}$ .

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Step 3.2.7.1

Set the denominator in $\frac{x + 1}{2 (3 - x)}$ equal to $0$ to find where the expression is undefined.

$2 (3 - x) = 0$

Step 3.2.7.2

Solve for $x$ .

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Step 3.2.7.2.1

Divide each term in $2 (3 - x) = 0$ by $2$ and simplify.

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Step 3.2.7.2.1.1

Divide each term in $2 (3 - x) = 0$ by $2$ .

$\frac{2 (3 - x)}{2} = \frac{0}{2}$

Step 3.2.7.2.1.2

Simplify the left side.

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Step 3.2.7.2.1.2.1

Cancel the common factor of $2$ .

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Step 3.2.7.2.1.2.1.1

Cancel the common factor.

$\frac{2 (3 - x)}{2} = \frac{0}{2}$

Step 3.2.7.2.1.2.1.2

Divide $3 - x$ by $1$ .

$3 - x = \frac{0}{2}$

Step 3.2.7.2.1.3

Simplify the right side.

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Step 3.2.7.2.1.3.1

Divide $0$ by $2$ .

$3 - x = 0$

Step 3.2.7.2.2

Subtract $3$ from both sides of the equation.

$- x = - 3$

Step 3.2.7.2.3

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

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Step 3.2.7.2.3.1

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

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

Step 3.2.7.2.3.2

Simplify the left side.

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Step 3.2.7.2.3.2.1

Dividing two negative values results in a positive value.

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

Step 3.2.7.2.3.2.2

Divide $x$ by $1$ .

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

Step 3.2.7.2.3.3

Simplify the right side.

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Step 3.2.7.2.3.3.1

Divide $- 3$ by $- 1$ .

$x = 3$

Step 3.2.7.3

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

$(- \infty, 3) \cup (3, \infty)$

Step 3.2.8

Use each root to create test intervals.

$x < - 1$

$- 1 < x < 3$

$x > 3$

Step 3.2.9

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 3.2.9.1

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

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Step 3.2.9.1.1

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

$x = - 4$

Step 3.2.9.1.2

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

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

Step 3.2.9.1.3

The left side $- 0.2\overline{142857}$ is not greater than the right side $0$ , which means that the given statement is false.

False

Step 3.2.9.2

Test a value on the interval $- 1 < x < 3$ to see if it makes the inequality true.

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Step 3.2.9.2.1

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

$x = 0$

Step 3.2.9.2.2

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

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

Step 3.2.9.2.3

The left side $0.1\overline{6}$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 3.2.9.3

Test a value on the interval $x > 3$ to see if it makes the inequality true.

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Step 3.2.9.3.1

Choose a value on the interval $x > 3$ and see if this value makes the original inequality true.

$x = 6$

Step 3.2.9.3.2

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

$\frac{(6) + 1}{2 (3 - (6))} > 0$

Step 3.2.9.3.3

The left side $- 1.1\overline{6}$ is not greater than the right side $0$ , which means that the given statement is false.

False

Step 3.2.9.4

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

$x < - 1$ False

$- 1 < x < 3$ True

$x > 3$ False

$x < - 1$ False

$- 1 < x < 3$ True

$x > 3$ False

Step 3.2.10

The solution consists of all of the true intervals.

$- 1 < x < 3$

Step 3.3

Set the denominator in $\frac{x + 1}{2 (3 - x)}$ equal to $0$ to find where the expression is undefined.

$2 (3 - x) = 0$

Step 3.4

Solve for $x$ .

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Step 3.4.1

Divide each term in $2 (3 - x) = 0$ by $2$ and simplify.

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Step 3.4.1.1

Divide each term in $2 (3 - x) = 0$ by $2$ .

$\frac{2 (3 - x)}{2} = \frac{0}{2}$

Step 3.4.1.2

Simplify the left side.

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Step 3.4.1.2.1

Cancel the common factor of $2$ .

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Step 3.4.1.2.1.1

Cancel the common factor.

$\frac{2 (3 - x)}{2} = \frac{0}{2}$

Step 3.4.1.2.1.2

Divide $3 - x$ by $1$ .

$3 - x = \frac{0}{2}$

Step 3.4.1.3

Simplify the right side.

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Step 3.4.1.3.1

Divide $0$ by $2$ .

$3 - x = 0$

Step 3.4.2

Subtract $3$ from both sides of the equation.

$- x = - 3$

Step 3.4.3

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

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Step 3.4.3.1

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

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

Step 3.4.3.2

Simplify the left side.

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Step 3.4.3.2.1

Dividing two negative values results in a positive value.

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

Step 3.4.3.2.2

Divide $x$ by $1$ .

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

Step 3.4.3.3

Simplify the right side.

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Step 3.4.3.3.1

Divide $- 3$ by $- 1$ .

$x = 3$

Step 3.5

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

$(- 1, 3)$

Step 4

Use each root to create test intervals.

$x < - 1$

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

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

$x > 3$

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 $x < - 1$ to see if it makes the inequality true.

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Step 5.1.1

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

$x = - 4$

Step 5.1.2

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

$\log_{2} ((- 4) + 1) < \log_{2} (6 - 2 \cdot - 4)$

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

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Step 5.2.1

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

$x = 0$

Step 5.2.2

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

$\log_{2} ((0) + 1) < \log_{2} (6 - 2 \cdot 0)$

Step 5.2.3

The left side $0$ is less than the right side $2.5849625$ , which means that the given statement is always true.

True

Step 5.3

Test a value on the interval $\frac{5}{3} < x < 3$ to see if it makes the inequality true.

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Step 5.3.1

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

$x = 2$

Step 5.3.2

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

$\log_{2} ((2) + 1) < \log_{2} (6 - 2 \cdot 2)$

Step 5.3.3

The left side $1.5849625$ 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 $x > 3$ to see if it makes the inequality true.

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Step 5.4.1

Choose a value on the interval $x > 3$ and see if this value makes the original inequality true.

$x = 6$

Step 5.4.2

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

$\log_{2} ((6) + 1) < \log_{2} (6 - 2 \cdot 6)$

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.

$x < - 1$ False

$- 1 < x < \frac{5}{3}$ True

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

$x > 3$ False

$x < - 1$ False

$- 1 < x < \frac{5}{3}$ True

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

$x > 3$ False

Step 6

The solution consists of all of the true intervals.

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

Step 7

The result can be shown in multiple forms.

Inequality Form:

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

Interval Notation:

$(- 1, \frac{5}{3})$

Step 8