Algebra Examples

$\frac{1}{6} (x - 15) < x + 20$

Step 1

Simplify $\frac{1}{6} \cdot (x - 15)$ .

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Step 1.1

Rewrite.

$0 + 0 + \frac{1}{6} \cdot (x - 15) < x + 20$

Step 1.2

Simplify by adding zeros.

$\frac{1}{6} \cdot (x - 15) < x + 20$

Step 1.3

Apply the distributive property.

$\frac{1}{6} x + \frac{1}{6} \cdot - 15 < x + 20$

Step 1.4

Combine $\frac{1}{6}$ and $x$ .

$\frac{x}{6} + \frac{1}{6} \cdot - 15 < x + 20$

Step 1.5

Cancel the common factor of $3$ .

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Step 1.5.1

Factor $3$ out of $6$ .

$\frac{x}{6} + \frac{1}{3 (2)} \cdot - 15 < x + 20$

Step 1.5.2

Factor $3$ out of $- 15$ .

$\frac{x}{6} + \frac{1}{3 \cdot 2} \cdot (3 \cdot - 5) < x + 20$

Step 1.5.3

Cancel the common factor.

$\frac{x}{6} + \frac{1}{3 \cdot 2} \cdot (3 \cdot - 5) < x + 20$

Step 1.5.4

Rewrite the expression.

$\frac{x}{6} + \frac{1}{2} \cdot - 5 < x + 20$

Step 1.6

Combine $\frac{1}{2}$ and $- 5$ .

$\frac{x}{6} + \frac{- 5}{2} < x + 20$

Step 1.7

Move the negative in front of the fraction.

$\frac{x}{6} - \frac{5}{2} < x + 20$

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.

$\frac{x}{6} - \frac{5}{2} - x < 20$

Step 2.2

To write $- x$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{x}{6} - x \cdot \frac{6}{6} - \frac{5}{2} < 20$

Step 2.3

Combine $- x$ and $\frac{6}{6}$ .

$\frac{x}{6} + \frac{- x \cdot 6}{6} - \frac{5}{2} < 20$

Step 2.4

Combine the numerators over the common denominator.

$\frac{x - x \cdot 6}{6} - \frac{5}{2} < 20$

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 $x - x \cdot 6$ .

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Step 2.5.1.1.1

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

$\frac{x^{1} - x \cdot 6}{6} - \frac{5}{2} < 20$

Step 2.5.1.1.2

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

$\frac{x \cdot 1 - x \cdot 6}{6} - \frac{5}{2} < 20$

Step 2.5.1.1.3

Factor $x$ out of $- x \cdot 6$ .

$\frac{x \cdot 1 + x (- 1 \cdot 6)}{6} - \frac{5}{2} < 20$

Step 2.5.1.1.4

Factor $x$ out of $x \cdot 1 + x (- 1 \cdot 6)$ .

$\frac{x (1 - 1 \cdot 6)}{6} - \frac{5}{2} < 20$

Step 2.5.1.2

Multiply $- 1$ by $6$ .

$\frac{x (1 - 6)}{6} - \frac{5}{2} < 20$

Step 2.5.1.3

Subtract $6$ from $1$ .

$\frac{x \cdot - 5}{6} - \frac{5}{2} < 20$

Step 2.5.2

Move $- 5$ to the left of $x$ .

$\frac{- 5 \cdot x}{6} - \frac{5}{2} < 20$

Step 2.5.3

Move the negative in front of the fraction.

$- \frac{5 x}{6} - \frac{5}{2} < 20$

Step 3

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

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Step 3.1

Add $\frac{5}{2}$ to both sides of the inequality.

$- \frac{5 x}{6} < 20 + \frac{5}{2}$

Step 3.2

To write $20$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{5 x}{6} < 20 \cdot \frac{2}{2} + \frac{5}{2}$

Step 3.3

Combine $20$ and $\frac{2}{2}$ .

$- \frac{5 x}{6} < \frac{20 \cdot 2}{2} + \frac{5}{2}$

Step 3.4

Combine the numerators over the common denominator.

$- \frac{5 x}{6} < \frac{20 \cdot 2 + 5}{2}$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Multiply $20$ by $2$ .

$- \frac{5 x}{6} < \frac{40 + 5}{2}$

Step 3.5.2

Add $40$ and $5$ .

$- \frac{5 x}{6} < \frac{45}{2}$

Step 4

Divide each term in $- \frac{5 x}{6} < \frac{45}{2}$ by $- 1$ and simplify.

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Step 4.1

Divide each term in $- \frac{5 x}{6} < \frac{45}{2}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{5 x}{6}}{- 1} > \frac{\frac{45}{2}}{- 1}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{5 x}{6}}{1} > \frac{\frac{45}{2}}{- 1}$

Step 4.2.2

Divide $\frac{5 x}{6}$ by $1$ .

$\frac{5 x}{6} > \frac{\frac{45}{2}}{- 1}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Move the negative one from the denominator of $\frac{\frac{45}{2}}{- 1}$ .

$\frac{5 x}{6} > - 1 \cdot \frac{45}{2}$

Step 4.3.2

Rewrite $- 1 \cdot \frac{45}{2}$ as $- \frac{45}{2}$ .

$\frac{5 x}{6} > - \frac{45}{2}$

Step 5

Multiply both sides by $6$ .

$\frac{5 x}{6} \cdot 6 > - \frac{45}{2} \cdot 6$

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 $6$ .

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Step 6.1.1.1

Cancel the common factor.

$\frac{5 x}{6} \cdot 6 > - \frac{45}{2} \cdot 6$

Step 6.1.1.2

Rewrite the expression.

$5 x > - \frac{45}{2} \cdot 6$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $- \frac{45}{2} \cdot 6$ .

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Step 6.2.1.1

Cancel the common factor of $2$ .

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Step 6.2.1.1.1

Move the leading negative in $- \frac{45}{2}$ into the numerator.

$5 x > \frac{- 45}{2} \cdot 6$

Step 6.2.1.1.2

Factor $2$ out of $6$ .

$5 x > \frac{- 45}{2} \cdot (2 (3))$

Step 6.2.1.1.3

Cancel the common factor.

$5 x > \frac{- 45}{2} \cdot (2 \cdot 3)$

Step 6.2.1.1.4

Rewrite the expression.

$5 x > - 45 \cdot 3$

Step 6.2.1.2

Multiply $- 45$ by $3$ .

$5 x > - 135$

Step 7

Divide each term in $5 x > - 135$ by $5$ and simplify.

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Step 7.1

Divide each term in $5 x > - 135$ by $5$ .

$\frac{5 x}{5} > \frac{- 135}{5}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $5$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{5 x}{5} > \frac{- 135}{5}$

Step 7.2.1.2

Divide $x$ by $1$ .

$x > \frac{- 135}{5}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Divide $- 135$ by $5$ .

$x > - 27$

Step 8

The result can be shown in multiple forms.

Inequality Form:

$x > - 27$

Interval Notation:

$(- 27, \infty)$