Trigonometry Examples

$\frac{2}{3} x - \frac{5}{6} > \frac{3}{4} x$

Step 1

Combine $\frac{2}{3}$ and $x$ .

$\frac{2 x}{3} - \frac{5}{6} > \frac{3}{4} x$

Step 2

Combine $\frac{3}{4}$ and $x$ .

$\frac{2 x}{3} - \frac{5}{6} > \frac{3 x}{4}$

Step 3

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

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Step 3.1

Subtract $\frac{3 x}{4}$ from both sides of the inequality.

$\frac{2 x}{3} - \frac{5}{6} - \frac{3 x}{4} > 0$

Step 3.2

To write $\frac{2 x}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{2 x}{3} \cdot \frac{4}{4} - \frac{3 x}{4} - \frac{5}{6} > 0$

Step 3.3

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

$\frac{2 x}{3} \cdot \frac{4}{4} - \frac{3 x}{4} \cdot \frac{3}{3} - \frac{5}{6} > 0$

Step 3.4

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.1

Multiply $\frac{2 x}{3}$ by $\frac{4}{4}$ .

$\frac{2 x \cdot 4}{3 \cdot 4} - \frac{3 x}{4} \cdot \frac{3}{3} - \frac{5}{6} > 0$

Step 3.4.2

Multiply $3$ by $4$ .

$\frac{2 x \cdot 4}{12} - \frac{3 x}{4} \cdot \frac{3}{3} - \frac{5}{6} > 0$

Step 3.4.3

Multiply $\frac{3 x}{4}$ by $\frac{3}{3}$ .

$\frac{2 x \cdot 4}{12} - \frac{3 x \cdot 3}{4 \cdot 3} - \frac{5}{6} > 0$

Step 3.4.4

Multiply $4$ by $3$ .

$\frac{2 x \cdot 4}{12} - \frac{3 x \cdot 3}{12} - \frac{5}{6} > 0$

Step 3.5

Combine the numerators over the common denominator.

$\frac{2 x \cdot 4 - 3 x \cdot 3}{12} - \frac{5}{6} > 0$

Step 3.6

Simplify each term.

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Step 3.6.1

Simplify the numerator.

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Step 3.6.1.1

Factor $x$ out of $2 x \cdot 4 - 3 x \cdot 3$ .

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Step 3.6.1.1.1

Factor $x$ out of $2 x \cdot 4$ .

$\frac{x (2 \cdot 4) - 3 x \cdot 3}{12} - \frac{5}{6} > 0$

Step 3.6.1.1.2

Factor $x$ out of $- 3 x \cdot 3$ .

$\frac{x (2 \cdot 4) + x (- 3 \cdot 3)}{12} - \frac{5}{6} > 0$

Step 3.6.1.1.3

Factor $x$ out of $x (2 \cdot 4) + x (- 3 \cdot 3)$ .

$\frac{x (2 \cdot 4 - 3 \cdot 3)}{12} - \frac{5}{6} > 0$

Step 3.6.1.2

Multiply $2$ by $4$ .

$\frac{x (8 - 3 \cdot 3)}{12} - \frac{5}{6} > 0$

Step 3.6.1.3

Multiply $- 3$ by $3$ .

$\frac{x (8 - 9)}{12} - \frac{5}{6} > 0$

Step 3.6.1.4

Subtract $9$ from $8$ .

$\frac{x \cdot - 1}{12} - \frac{5}{6} > 0$

Step 3.6.2

Move $- 1$ to the left of $x$ .

$\frac{- 1 \cdot x}{12} - \frac{5}{6} > 0$

Step 3.6.3

Move the negative in front of the fraction.

$- \frac{x}{12} - \frac{5}{6} > 0$

Step 4

Add $\frac{5}{6}$ to both sides of the inequality.

$- \frac{x}{12} > \frac{5}{6}$

Step 5

Divide each term in $- \frac{x}{12} > \frac{5}{6}$ by $- 1$ and simplify.

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Step 5.1

Divide each term in $- \frac{x}{12} > \frac{5}{6}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{x}{12}}{- 1} < \frac{\frac{5}{6}}{- 1}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{x}{12}}{1} < \frac{\frac{5}{6}}{- 1}$

Step 5.2.2

Divide $\frac{x}{12}$ by $1$ .

$\frac{x}{12} < \frac{\frac{5}{6}}{- 1}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Move the negative one from the denominator of $\frac{\frac{5}{6}}{- 1}$ .

$\frac{x}{12} < - 1 \cdot \frac{5}{6}$

Step 5.3.2

Rewrite $- 1 \cdot \frac{5}{6}$ as $- \frac{5}{6}$ .

$\frac{x}{12} < - \frac{5}{6}$

Step 6

Multiply both sides by $12$ .

$\frac{x}{12} \cdot 12 < - \frac{5}{6} \cdot 12$

Step 7

Simplify.

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Step 7.1

Simplify the left side.

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Step 7.1.1

Cancel the common factor of $12$ .

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Step 7.1.1.1

Cancel the common factor.

$\frac{x}{12} \cdot 12 < - \frac{5}{6} \cdot 12$

Step 7.1.1.2

Rewrite the expression.

$x < - \frac{5}{6} \cdot 12$

Step 7.2

Simplify the right side.

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Step 7.2.1

Simplify $- \frac{5}{6} \cdot 12$ .

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Step 7.2.1.1

Cancel the common factor of $6$ .

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Step 7.2.1.1.1

Move the leading negative in $- \frac{5}{6}$ into the numerator.

$x < \frac{- 5}{6} \cdot 12$

Step 7.2.1.1.2

Factor $6$ out of $12$ .

$x < \frac{- 5}{6} \cdot (6 (2))$

Step 7.2.1.1.3

Cancel the common factor.

$x < \frac{- 5}{6} \cdot (6 \cdot 2)$

Step 7.2.1.1.4

Rewrite the expression.

$x < - 5 \cdot 2$

Step 7.2.1.2

Multiply $- 5$ by $2$ .

$x < - 10$

Step 8

The result can be shown in multiple forms.

Inequality Form:

$x < - 10$

Interval Notation:

$(- \infty, - 10)$