Algebra Examples

$\frac{1}{5} (x - 12) > x + 16$

Step 1

Solve $x < - 23$ .

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Step 1.1

Simplify $\frac{1}{5} (x - 12)$ .

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Step 1.1.1

Rewrite.

$0 + 0 + \frac{1}{5} (x - 12) > x + 16$

Step 1.1.2

Simplify by adding zeros.

$\frac{1}{5} (x - 12) > x + 16$

Step 1.1.3

Apply the distributive property.

$\frac{1}{5} x + \frac{1}{5} \cdot - 12 > x + 16$

Step 1.1.4

Combine $\frac{1}{5}$ and $x$ .

$\frac{x}{5} + \frac{1}{5} \cdot - 12 > x + 16$

Step 1.1.5

Combine $\frac{1}{5}$ and $- 12$ .

$\frac{x}{5} + \frac{- 12}{5} > x + 16$

Step 1.1.6

Move the negative in front of the fraction.

$\frac{x}{5} - \frac{12}{5} > x + 16$

Step 1.2

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

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Step 1.2.1

Subtract $x$ from both sides of the inequality.

$\frac{x}{5} - \frac{12}{5} - x > 16$

Step 1.2.2

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

$\frac{x}{5} - x \cdot \frac{5}{5} - \frac{12}{5} > 16$

Step 1.2.3

Combine $- x$ and $\frac{5}{5}$ .

$\frac{x}{5} + \frac{- x \cdot 5}{5} - \frac{12}{5} > 16$

Step 1.2.4

Combine the numerators over the common denominator.

$\frac{x - x \cdot 5}{5} - \frac{12}{5} > 16$

Step 1.2.5

Combine the numerators over the common denominator.

$\frac{x - x \cdot 5 - 12}{5} > 16$

Step 1.2.6

Multiply $5$ by $- 1$ .

$\frac{x - 5 x - 12}{5} > 16$

Step 1.2.7

Subtract $5 x$ from $x$ .

$\frac{- 4 x - 12}{5} > 16$

Step 1.2.8

Factor $4$ out of $- 4 x - 12$ .

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Step 1.2.8.1

Factor $4$ out of $- 4 x$ .

$\frac{4 (- x) - 12}{5} > 16$

Step 1.2.8.2

Factor $4$ out of $- 12$ .

$\frac{4 (- x) + 4 (- 3)}{5} > 16$

Step 1.2.8.3

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

$\frac{4 (- x - 3)}{5} > 16$

Step 1.2.9

Factor $- 1$ out of $- x$ .

$\frac{4 (- (x) - 3)}{5} > 16$

Step 1.2.10

Rewrite $- 3$ as $- 1 (3)$ .

$\frac{4 (- (x) - 1 (3))}{5} > 16$

Step 1.2.11

Factor $- 1$ out of $- (x) - 1 (3)$ .

$\frac{4 (- (x + 3))}{5} > 16$

Step 1.2.12

Rewrite $- (x + 3)$ as $- 1 (x + 3)$ .

$\frac{4 (- 1 (x + 3))}{5} > 16$

Step 1.2.13

Move the negative in front of the fraction.

$- \frac{4 (x + 3)}{5} > 16$

Step 1.3

Divide each term in $- \frac{4 (x + 3)}{5} > 16$ by $- 1$ and simplify.

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Step 1.3.1

Divide each term in $- \frac{4 (x + 3)}{5} > 16$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{4 (x + 3)}{5}}{- 1} < \frac{16}{- 1}$

Step 1.3.2

Simplify the left side.

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Step 1.3.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{4 (x + 3)}{5}}{1} < \frac{16}{- 1}$

Step 1.3.2.2

Divide $\frac{4 (x + 3)}{5}$ by $1$ .

$\frac{4 (x + 3)}{5} < \frac{16}{- 1}$

Step 1.3.3

Simplify the right side.

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Step 1.3.3.1

Divide $16$ by $- 1$ .

$\frac{4 (x + 3)}{5} < - 16$

Step 1.4

Multiply both sides by $5$ .

$\frac{4 (x + 3)}{5} \cdot 5 < - 16 \cdot 5$

Step 1.5

Simplify.

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Step 1.5.1

Simplify the left side.

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Step 1.5.1.1

Simplify $\frac{4 (x + 3)}{5} \cdot 5$ .

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Step 1.5.1.1.1

Cancel the common factor of $5$ .

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Step 1.5.1.1.1.1

Cancel the common factor.

$\frac{4 (x + 3)}{5} \cdot 5 < - 16 \cdot 5$

Step 1.5.1.1.1.2

Rewrite the expression.

$4 (x + 3) < - 16 \cdot 5$

Step 1.5.1.1.2

Apply the distributive property.

$4 x + 4 \cdot 3 < - 16 \cdot 5$

Step 1.5.1.1.3

Multiply $4$ by $3$ .

$4 x + 12 < - 16 \cdot 5$

Step 1.5.2

Simplify the right side.

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Step 1.5.2.1

Multiply $- 16$ by $5$ .

$4 x + 12 < - 80$

Step 1.6

Solve for $x$ .

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Step 1.6.1

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

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Step 1.6.1.1

Subtract $12$ from both sides of the inequality.

$4 x < - 80 - 12$

Step 1.6.1.2

Subtract $12$ from $- 80$ .

$4 x < - 92$

Step 1.6.2

Divide each term in $4 x < - 92$ by $4$ and simplify.

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Step 1.6.2.1

Divide each term in $4 x < - 92$ by $4$ .

$\frac{4 x}{4} < \frac{- 92}{4}$

Step 1.6.2.2

Simplify the left side.

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Step 1.6.2.2.1

Cancel the common factor of $4$ .

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Step 1.6.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} < \frac{- 92}{4}$

Step 1.6.2.2.1.2

Divide $x$ by $1$ .

$x < \frac{- 92}{4}$

Step 1.6.2.3

Simplify the right side.

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Step 1.6.2.3.1

Divide $- 92$ by $4$ .

$x < - 23$

Step 2

Use the inequality $x < - 23$ to build the set notation.

${x | x < - 23}$

Step 3