Trigonometry Examples

$\frac{6}{3 w - 4} > w + 1$

Step 1

Move all the expressions to the left side of the equation.

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Step 1.1

Subtract $w$ from both sides of the inequality.

$\frac{6}{3 w - 4} - w > 1$

Step 1.2

Subtract $1$ from both sides of the inequality.

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

Step 2

Simplify $\frac{6}{3 w - 4} - w - 1$ .

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Step 2.1

To write $- w$ as a fraction with a common denominator, multiply by $\frac{3 w - 4}{3 w - 4}$ .

$\frac{6}{3 w - 4} - w \cdot \frac{3 w - 4}{3 w - 4} - 1 > 0$

Step 2.2

Combine $- w$ and $\frac{3 w - 4}{3 w - 4}$ .

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

Step 2.3

Combine the numerators over the common denominator.

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

Step 2.4

Simplify the numerator.

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Step 2.4.1

Apply the distributive property.

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

Step 2.4.2

Rewrite using the commutative property of multiplication.

$\frac{6 - 1 \cdot 3 w \cdot w - w \cdot - 4}{3 w - 4} - 1 > 0$

Step 2.4.3

Multiply $- 4$ by $- 1$ .

$\frac{6 - 1 \cdot 3 w \cdot w + 4 w}{3 w - 4} - 1 > 0$

Step 2.4.4

Simplify each term.

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Step 2.4.4.1

Multiply $w$ by $w$ by adding the exponents.

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Step 2.4.4.1.1

Move $w$ .

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

Step 2.4.4.1.2

Multiply $w$ by $w$ .

$\frac{6 - 1 \cdot 3 w^{2} + 4 w}{3 w - 4} - 1 > 0$

Step 2.4.4.2

Multiply $- 1$ by $3$ .

$\frac{6 - 3 w^{2} + 4 w}{3 w - 4} - 1 > 0$

Step 2.4.5

Reorder terms.

$\frac{- 3 w^{2} + 4 w + 6}{3 w - 4} - 1 > 0$

Step 2.5

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3 w - 4}{3 w - 4}$ .

$\frac{- 3 w^{2} + 4 w + 6}{3 w - 4} - 1 \cdot \frac{3 w - 4}{3 w - 4} > 0$

Step 2.6

Combine $- 1$ and $\frac{3 w - 4}{3 w - 4}$ .

$\frac{- 3 w^{2} + 4 w + 6}{3 w - 4} + \frac{- (3 w - 4)}{3 w - 4} > 0$

Step 2.7

Combine the numerators over the common denominator.

$\frac{- 3 w^{2} + 4 w + 6 - (3 w - 4)}{3 w - 4} > 0$

Step 2.8

Simplify the numerator.

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Step 2.8.1

Apply the distributive property.

$\frac{- 3 w^{2} + 4 w + 6 - (3 w) - - 4}{3 w - 4} > 0$

Step 2.8.2

Multiply $3$ by $- 1$ .

$\frac{- 3 w^{2} + 4 w + 6 - 3 w - - 4}{3 w - 4} > 0$

Step 2.8.3

Multiply $- 1$ by $- 4$ .

$\frac{- 3 w^{2} + 4 w + 6 - 3 w + 4}{3 w - 4} > 0$

Step 2.8.4

Subtract $3 w$ from $4 w$ .

$\frac{- 3 w^{2} + w + 6 + 4}{3 w - 4} > 0$

Step 2.8.5

Add $6$ and $4$ .

$\frac{- 3 w^{2} + w + 10}{3 w - 4} > 0$

Step 2.8.6

Factor by grouping.

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Step 2.8.6.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 3 \cdot 10 = - 30$ and whose sum is $b = 1$ .

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Step 2.8.6.1.1

Multiply by $1$ .

$\frac{- 3 w^{2} + 1 w + 10}{3 w - 4} > 0$

Step 2.8.6.1.2

Rewrite $1$ as $- 5$ plus $6$

$\frac{- 3 w^{2} + (- 5 + 6) w + 10}{3 w - 4} > 0$

Step 2.8.6.1.3

Apply the distributive property.

$\frac{- 3 w^{2} - 5 w + 6 w + 10}{3 w - 4} > 0$

Step 2.8.6.2

Factor out the greatest common factor from each group.

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Step 2.8.6.2.1

Group the first two terms and the last two terms.

$\frac{(- 3 w^{2} - 5 w) + 6 w + 10}{3 w - 4} > 0$

Step 2.8.6.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{w (- 3 w - 5) - 2 (- 3 w - 5)}{3 w - 4} > 0$

Step 2.8.6.3

Factor the polynomial by factoring out the greatest common factor, $- 3 w - 5$ .

$\frac{(- 3 w - 5) (w - 2)}{3 w - 4} > 0$

Step 2.9

Factor $- 1$ out of $- 3 w$ .

$\frac{(- (3 w) - 5) (w - 2)}{3 w - 4} > 0$

Step 2.10

Rewrite $- 5$ as $- 1 (5)$ .

$\frac{(- (3 w) - 1 (5)) (w - 2)}{3 w - 4} > 0$

Step 2.11

Factor $- 1$ out of $- (3 w) - 1 (5)$ .

$\frac{- (3 w + 5) (w - 2)}{3 w - 4} > 0$

Step 2.12

Rewrite $- (3 w + 5)$ as $- 1 (3 w + 5)$ .

$\frac{- 1 (3 w + 5) (w - 2)}{3 w - 4} > 0$

Step 2.13

Move the negative in front of the fraction.

$- \frac{(3 w + 5) (w - 2)}{3 w - 4} > 0$

Step 3

Find all the values where the expression switches from negative to positive by setting each factor equal to $0$ and solving.

$3 w + 5 = 0$

$w - 2 = 0$

$3 w - 4 = 0$

Step 4

Subtract $5$ from both sides of the equation.

$3 w = - 5$

Step 5

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

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Step 5.1

Divide each term in $3 w = - 5$ by $3$ .

$\frac{3 w}{3} = \frac{- 5}{3}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $3$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{3 w}{3} = \frac{- 5}{3}$

Step 5.2.1.2

Divide $w$ by $1$ .

$w = \frac{- 5}{3}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Move the negative in front of the fraction.

$w = - \frac{5}{3}$

Step 6

Add $2$ to both sides of the equation.

$w = 2$

Step 7

Add $4$ to both sides of the equation.

$3 w = 4$

Step 8

Divide each term in $3 w = 4$ by $3$ and simplify.

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Step 8.1

Divide each term in $3 w = 4$ by $3$ .

$\frac{3 w}{3} = \frac{4}{3}$

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $3$ .

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Step 8.2.1.1

Cancel the common factor.

$\frac{3 w}{3} = \frac{4}{3}$

Step 8.2.1.2

Divide $w$ by $1$ .

$w = \frac{4}{3}$

Step 9

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

$w = - \frac{5}{3}$

$w = 2$

$w = \frac{4}{3}$

Step 10

Consolidate the solutions.

$w = - \frac{5}{3}, 2, \frac{4}{3}$

Step 11

Find the domain of $- \frac{(3 w + 5) (w - 2)}{3 w - 4}$ .

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Step 11.1

Set the denominator in $\frac{(3 w + 5) (w - 2)}{3 w - 4}$ equal to $0$ to find where the expression is undefined.

$3 w - 4 = 0$

Step 11.2

Solve for $w$ .

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Step 11.2.1

Add $4$ to both sides of the equation.

$3 w = 4$

Step 11.2.2

Divide each term in $3 w = 4$ by $3$ and simplify.

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Step 11.2.2.1

Divide each term in $3 w = 4$ by $3$ .

$\frac{3 w}{3} = \frac{4}{3}$

Step 11.2.2.2

Simplify the left side.

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Step 11.2.2.2.1

Cancel the common factor of $3$ .

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Step 11.2.2.2.1.1

Cancel the common factor.

$\frac{3 w}{3} = \frac{4}{3}$

Step 11.2.2.2.1.2

Divide $w$ by $1$ .

$w = \frac{4}{3}$

Step 11.3

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

$(- \infty, \frac{4}{3}) \cup (\frac{4}{3}, \infty)$

Step 12

Use each root to create test intervals.

$w < - \frac{5}{3}$

$- \frac{5}{3} < w < \frac{4}{3}$

$\frac{4}{3} < w < 2$

$w > 2$

Step 13

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 13.1

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

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Step 13.1.1

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

$w = - 4$

Step 13.1.2

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

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

Step 13.1.3

The left side $- 0.375$ is greater than the right side $- 3$ , which means that the given statement is always true.

True

Step 13.2

Test a value on the interval $- \frac{5}{3} < w < \frac{4}{3}$ to see if it makes the inequality true.

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Step 13.2.1

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

$w = 0$

Step 13.2.2

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

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

Step 13.2.3

The left side $- 1.5$ is not greater than the right side $1$ , which means that the given statement is false.

False

Step 13.3

Test a value on the interval $\frac{4}{3} < w < 2$ to see if it makes the inequality true.

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Step 13.3.1

Choose a value on the interval $\frac{4}{3} < w < 2$ and see if this value makes the original inequality true.

$w = 1.67$

Step 13.3.2

Replace $w$ with $1.67$ in the original inequality.

$\frac{6}{3 (1.67) - 4} > (1.67) + 1$

Step 13.3.3

The left side $5.\overline{9405}$ is greater than the right side $2.67$ , which means that the given statement is always true.

True

Step 13.4

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

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Step 13.4.1

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

$w = 4$

Step 13.4.2

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

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

Step 13.4.3

The left side $0.75$ is not greater than the right side $5$ , which means that the given statement is false.

False

Step 13.5

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

$w < - \frac{5}{3}$ True

$- \frac{5}{3} < w < \frac{4}{3}$ False

$\frac{4}{3} < w < 2$ True

$w > 2$ False

$w < - \frac{5}{3}$ True

$- \frac{5}{3} < w < \frac{4}{3}$ False

$\frac{4}{3} < w < 2$ True

$w > 2$ False

Step 14

The solution consists of all of the true intervals.

$w < - \frac{5}{3}$ or $\frac{4}{3} < w < 2$

Step 15

Convert the inequality to interval notation.

$(- \infty, - \frac{5}{3}) \cup (\frac{4}{3}, 2)$

Step 16