Trigonometry Examples

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

Step 1

Simplify $\frac{1}{2} \cdot (x + 3) + 2 (x - 4)$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

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

Step 1.1.2

Combine $\frac{1}{2}$ and $x$ .

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

Step 1.1.3

Combine $\frac{1}{2}$ and $3$ .

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

Step 1.1.4

Apply the distributive property.

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

Step 1.1.5

Multiply $2$ by $- 4$ .

$\frac{x}{2} + \frac{3}{2} + 2 x - 8 > \frac{1}{3} \cdot (x - 3)$

Step 1.2

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

$\frac{x}{2} + 2 x \cdot \frac{2}{2} + \frac{3}{2} - 8 > \frac{1}{3} \cdot (x - 3)$

Step 1.3

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

$\frac{x}{2} + \frac{2 x \cdot 2}{2} + \frac{3}{2} - 8 > \frac{1}{3} \cdot (x - 3)$

Step 1.4

Combine the numerators over the common denominator.

$\frac{x + 2 x \cdot 2}{2} + \frac{3}{2} - 8 > \frac{1}{3} \cdot (x - 3)$

Step 1.5

Combine the numerators over the common denominator.

$- 8 + \frac{x + 2 x \cdot 2 + 3}{2} > \frac{1}{3} \cdot (x - 3)$

Step 1.6

Multiply $2$ by $2$ .

$- 8 + \frac{x + 4 x + 3}{2} > \frac{1}{3} \cdot (x - 3)$

Step 1.7

Add $x$ and $4 x$ .

$- 8 + \frac{5 x + 3}{2} > \frac{1}{3} \cdot (x - 3)$

Step 1.8

To write $- 8$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- 8 \cdot \frac{2}{2} + \frac{5 x + 3}{2} > \frac{1}{3} \cdot (x - 3)$

Step 1.9

Simplify terms.

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Step 1.9.1

Combine $- 8$ and $\frac{2}{2}$ .

$\frac{- 8 \cdot 2}{2} + \frac{5 x + 3}{2} > \frac{1}{3} \cdot (x - 3)$

Step 1.9.2

Combine the numerators over the common denominator.

$\frac{- 8 \cdot 2 + 5 x + 3}{2} > \frac{1}{3} \cdot (x - 3)$

Step 1.10

Simplify the numerator.

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Step 1.10.1

Multiply $- 8$ by $2$ .

$\frac{- 16 + 5 x + 3}{2} > \frac{1}{3} \cdot (x - 3)$

Step 1.10.2

Add $- 16$ and $3$ .

$\frac{5 x - 13}{2} > \frac{1}{3} \cdot (x - 3)$

Step 2

Simplify $\frac{1}{3} \cdot (x - 3)$ .

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Step 2.1

Apply the distributive property.

$\frac{5 x - 13}{2} > \frac{1}{3} x + \frac{1}{3} \cdot - 3$

Step 2.2

Combine $\frac{1}{3}$ and $x$ .

$\frac{5 x - 13}{2} > \frac{x}{3} + \frac{1}{3} \cdot - 3$

Step 2.3

Cancel the common factor of $3$ .

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Step 2.3.1

Factor $3$ out of $- 3$ .

$\frac{5 x - 13}{2} > \frac{x}{3} + \frac{1}{3} \cdot (3 (- 1))$

Step 2.3.2

Cancel the common factor.

$\frac{5 x - 13}{2} > \frac{x}{3} + \frac{1}{3} \cdot (3 \cdot - 1)$

Step 2.3.3

Rewrite the expression.

$\frac{5 x - 13}{2} > \frac{x}{3} - 1$

Step 3

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

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Step 3.1

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

$\frac{5 x - 13}{2} - \frac{x}{3} > - 1$

Step 3.2

To write $\frac{5 x - 13}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{5 x - 13}{2} \cdot \frac{3}{3} - \frac{x}{3} > - 1$

Step 3.3

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

$\frac{5 x - 13}{2} \cdot \frac{3}{3} - \frac{x}{3} \cdot \frac{2}{2} > - 1$

Step 3.4

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.1

Multiply $\frac{5 x - 13}{2}$ by $\frac{3}{3}$ .

$\frac{(5 x - 13) \cdot 3}{2 \cdot 3} - \frac{x}{3} \cdot \frac{2}{2} > - 1$

Step 3.4.2

Multiply $2$ by $3$ .

$\frac{(5 x - 13) \cdot 3}{6} - \frac{x}{3} \cdot \frac{2}{2} > - 1$

Step 3.4.3

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

$\frac{(5 x - 13) \cdot 3}{6} - \frac{x \cdot 2}{3 \cdot 2} > - 1$

Step 3.4.4

Multiply $3$ by $2$ .

$\frac{(5 x - 13) \cdot 3}{6} - \frac{x \cdot 2}{6} > - 1$

Step 3.5

Combine the numerators over the common denominator.

$\frac{(5 x - 13) \cdot 3 - x \cdot 2}{6} > - 1$

Step 3.6

Simplify the numerator.

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Step 3.6.1

Apply the distributive property.

$\frac{5 x \cdot 3 - 13 \cdot 3 - x \cdot 2}{6} > - 1$

Step 3.6.2

Multiply $3$ by $5$ .

$\frac{15 x - 13 \cdot 3 - x \cdot 2}{6} > - 1$

Step 3.6.3

Multiply $- 13$ by $3$ .

$\frac{15 x - 39 - x \cdot 2}{6} > - 1$

Step 3.6.4

Multiply $2$ by $- 1$ .

$\frac{15 x - 39 - 2 x}{6} > - 1$

Step 3.6.5

Subtract $2 x$ from $15 x$ .

$\frac{13 x - 39}{6} > - 1$

Step 3.6.6

Factor $13$ out of $13 x - 39$ .

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Step 3.6.6.1

Factor $13$ out of $13 x$ .

$\frac{13 (x) - 39}{6} > - 1$

Step 3.6.6.2

Factor $13$ out of $- 39$ .

$\frac{13 x + 13 \cdot - 3}{6} > - 1$

Step 3.6.6.3

Factor $13$ out of $13 x + 13 \cdot - 3$ .

$\frac{13 (x - 3)}{6} > - 1$

Step 4

Multiply both sides by $6$ .

$\frac{13 (x - 3)}{6} \cdot 6 > - 1 \cdot 6$

Step 5

Simplify.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Simplify $\frac{13 (x - 3)}{6} \cdot 6$ .

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Step 5.1.1.1

Cancel the common factor of $6$ .

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Step 5.1.1.1.1

Cancel the common factor.

$\frac{13 (x - 3)}{6} \cdot 6 > - 1 \cdot 6$

Step 5.1.1.1.2

Rewrite the expression.

$13 (x - 3) > - 1 \cdot 6$

Step 5.1.1.2

Apply the distributive property.

$13 x + 13 \cdot - 3 > - 1 \cdot 6$

Step 5.1.1.3

Multiply $13$ by $- 3$ .

$13 x - 39 > - 1 \cdot 6$

Step 5.2

Simplify the right side.

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Step 5.2.1

Multiply $- 1$ by $6$ .

$13 x - 39 > - 6$

Step 6

Solve for $x$ .

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Step 6.1

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

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Step 6.1.1

Add $39$ to both sides of the inequality.

$13 x > - 6 + 39$

Step 6.1.2

Add $- 6$ and $39$ .

$13 x > 33$

Step 6.2

Divide each term in $13 x > 33$ by $13$ and simplify.

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Step 6.2.1

Divide each term in $13 x > 33$ by $13$ .

$\frac{13 x}{13} > \frac{33}{13}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Cancel the common factor of $13$ .

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Step 6.2.2.1.1

Cancel the common factor.

$\frac{13 x}{13} > \frac{33}{13}$

Step 6.2.2.1.2

Divide $x$ by $1$ .

$x > \frac{33}{13}$

Step 7

The result can be shown in multiple forms.

Inequality Form:

$x > \frac{33}{13}$

Interval Notation:

$(\frac{33}{13}, \infty)$