Finite Math Examples

$\frac{1}{3} \cdot (x - 6) > x + 8$

Step 1

Simplify $\frac{1}{3} \cdot (x - 6)$ .

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Step 1.1

Rewrite.

$0 + 0 + \frac{1}{3} \cdot (x - 6) > x + 8$

Step 1.2

Simplify by adding zeros.

$\frac{1}{3} \cdot (x - 6) > x + 8$

Step 1.3

Apply the distributive property.

$\frac{1}{3} x + \frac{1}{3} \cdot - 6 > x + 8$

Step 1.4

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

$\frac{x}{3} + \frac{1}{3} \cdot - 6 > x + 8$

Step 1.5

Cancel the common factor of $3$ .

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Step 1.5.1

Factor $3$ out of $- 6$ .

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

Step 1.5.2

Cancel the common factor.

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

Step 1.5.3

Rewrite the expression.

$\frac{x}{3} - 2 > x + 8$

Step 2

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

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Step 2.1

Subtract $x$ from both sides of the inequality.

$\frac{x}{3} - 2 - x > 8$

Step 2.2

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

$\frac{x}{3} - x \cdot \frac{3}{3} - 2 > 8$

Step 2.3

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

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

Step 2.4

Combine the numerators over the common denominator.

$\frac{x - x \cdot 3}{3} - 2 > 8$

Step 2.5

Simplify each term.

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Step 2.5.1

Simplify the numerator.

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Step 2.5.1.1

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

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Step 2.5.1.1.1

Raise $x$ to the power of $1$ .

$\frac{x^{1} - x \cdot 3}{3} - 2 > 8$

Step 2.5.1.1.2

Factor $x$ out of $x^{1}$ .

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

Step 2.5.1.1.3

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

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

Step 2.5.1.1.4

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

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

Step 2.5.1.2

Multiply $- 1$ by $3$ .

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

Step 2.5.1.3

Subtract $3$ from $1$ .

$\frac{x \cdot - 2}{3} - 2 > 8$

Step 2.5.2

Move $- 2$ to the left of $x$ .

$\frac{- 2 \cdot x}{3} - 2 > 8$

Step 2.5.3

Move the negative in front of the fraction.

$- \frac{2 x}{3} - 2 > 8$

Step 3

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

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Step 3.1

Add $2$ to both sides of the inequality.

$- \frac{2 x}{3} > 8 + 2$

Step 3.2

Add $8$ and $2$ .

$- \frac{2 x}{3} > 10$

Step 4

Divide each term in $- \frac{2 x}{3} > 10$ by $- 1$ and simplify.

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Step 4.1

Divide each term in $- \frac{2 x}{3} > 10$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{2 x}{3}}{- 1} < \frac{10}{- 1}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{2 x}{3}}{1} < \frac{10}{- 1}$

Step 4.2.2

Divide $\frac{2 x}{3}$ by $1$ .

$\frac{2 x}{3} < \frac{10}{- 1}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $10$ by $- 1$ .

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

Step 5

Multiply both sides by $3$ .

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

Step 6

Simplify.

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Step 6.1

Simplify the left side.

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Step 6.1.1

Cancel the common factor of $3$ .

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Step 6.1.1.1

Cancel the common factor.

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

Step 6.1.1.2

Rewrite the expression.

$2 x < - 10 \cdot 3$

Step 6.2

Simplify the right side.

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Step 6.2.1

Multiply $- 10$ by $3$ .

$2 x < - 30$

Step 7

Divide each term in $2 x < - 30$ by $2$ and simplify.

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Step 7.1

Divide each term in $2 x < - 30$ by $2$ .

$\frac{2 x}{2} < \frac{- 30}{2}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $2$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{2 x}{2} < \frac{- 30}{2}$

Step 7.2.1.2

Divide $x$ by $1$ .

$x < \frac{- 30}{2}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Divide $- 30$ by $2$ .

$x < - 15$

Step 8

Convert the inequality to interval notation.

$(- \infty, - 15)$

Step 9