Trigonometry Examples

$\frac{9}{2} v + 6 < 3 v - 4$

Step 1

Combine $\frac{9}{2}$ and $v$ .

$\frac{9 v}{2} + 6 < 3 v - 4$

Step 2

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

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Step 2.1

Subtract $3 v$ from both sides of the inequality.

$\frac{9 v}{2} + 6 - 3 v < - 4$

Step 2.2

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

$\frac{9 v}{2} - 3 v \cdot \frac{2}{2} + 6 < - 4$

Step 2.3

Combine $- 3 v$ and $\frac{2}{2}$ .

$\frac{9 v}{2} + \frac{- 3 v \cdot 2}{2} + 6 < - 4$

Step 2.4

Combine the numerators over the common denominator.

$\frac{9 v - 3 v \cdot 2}{2} + 6 < - 4$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Factor $3 v$ out of $9 v - 3 v \cdot 2$ .

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Step 2.5.1.1

Factor $3 v$ out of $9 v$ .

$\frac{3 v (3) - 3 v \cdot 2}{2} + 6 < - 4$

Step 2.5.1.2

Factor $3 v$ out of $- 3 v \cdot 2$ .

$\frac{3 v (3) + 3 v (- 1 \cdot 2)}{2} + 6 < - 4$

Step 2.5.1.3

Factor $3 v$ out of $3 v (3) + 3 v (- 1 \cdot 2)$ .

$\frac{3 v (3 - 1 \cdot 2)}{2} + 6 < - 4$

Step 2.5.2

Multiply $- 1$ by $2$ .

$\frac{3 v (3 - 2)}{2} + 6 < - 4$

Step 2.5.3

Subtract $2$ from $3$ .

$\frac{3 v \cdot 1}{2} + 6 < - 4$

Step 2.5.4

Multiply $3$ by $1$ .

$\frac{3 v}{2} + 6 < - 4$

Step 3

Move all terms not containing $v$ to the right side of the inequality.

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Step 3.1

Subtract $6$ from both sides of the inequality.

$\frac{3 v}{2} < - 4 - 6$

Step 3.2

Subtract $6$ from $- 4$ .

$\frac{3 v}{2} < - 10$

Step 4

Multiply both sides by $2$ .

$\frac{3 v}{2} \cdot 2 < - 10 \cdot 2$

Step 5

Simplify.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Cancel the common factor of $2$ .

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Step 5.1.1.1

Cancel the common factor.

$\frac{3 v}{2} \cdot 2 < - 10 \cdot 2$

Step 5.1.1.2

Rewrite the expression.

$3 v < - 10 \cdot 2$

Step 5.2

Simplify the right side.

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Step 5.2.1

Multiply $- 10$ by $2$ .

$3 v < - 20$

Step 6

Divide each term in $3 v < - 20$ by $3$ and simplify.

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Step 6.1

Divide each term in $3 v < - 20$ by $3$ .

$\frac{3 v}{3} < \frac{- 20}{3}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $3$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{3 v}{3} < \frac{- 20}{3}$

Step 6.2.1.2

Divide $v$ by $1$ .

$v < \frac{- 20}{3}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Move the negative in front of the fraction.

$v < - \frac{20}{3}$

Step 7

The result can be shown in multiple forms.

Inequality Form:

$v < - \frac{20}{3}$

Interval Notation:

$(- \infty, - \frac{20}{3})$