Algebra Examples

$\frac{1}{4} (x - 9) < x + 12$

Step 1

Simplify $\frac{1}{4} \cdot (x - 9)$ .

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Step 1.1

Rewrite.

$0 + 0 + \frac{1}{4} \cdot (x - 9) < x + 12$

Step 1.2

Simplify by adding zeros.

$\frac{1}{4} \cdot (x - 9) < x + 12$

Step 1.3

Apply the distributive property.

$\frac{1}{4} x + \frac{1}{4} \cdot - 9 < x + 12$

Step 1.4

Combine $\frac{1}{4}$ and $x$ .

$\frac{x}{4} + \frac{1}{4} \cdot - 9 < x + 12$

Step 1.5

Combine $\frac{1}{4}$ and $- 9$ .

$\frac{x}{4} + \frac{- 9}{4} < x + 12$

Step 1.6

Move the negative in front of the fraction.

$\frac{x}{4} - \frac{9}{4} < x + 12$

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.

$\frac{x}{4} - \frac{9}{4} - x < 12$

Step 2.2

To write $- x$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{x}{4} - x \cdot \frac{4}{4} - \frac{9}{4} < 12$

Step 2.3

Combine $- x$ and $\frac{4}{4}$ .

$\frac{x}{4} + \frac{- x \cdot 4}{4} - \frac{9}{4} < 12$

Step 2.4

Combine the numerators over the common denominator.

$\frac{x - x \cdot 4}{4} - \frac{9}{4} < 12$

Step 2.5

Combine the numerators over the common denominator.

$\frac{x - x \cdot 4 - 9}{4} < 12$

Step 2.6

Multiply $4$ by $- 1$ .

$\frac{x - 4 x - 9}{4} < 12$

Step 2.7

Subtract $4 x$ from $x$ .

$\frac{- 3 x - 9}{4} < 12$

Step 2.8

Factor $3$ out of $- 3 x - 9$ .

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Step 2.8.1

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

$\frac{3 (- x) - 9}{4} < 12$

Step 2.8.2

Factor $3$ out of $- 9$ .

$\frac{3 (- x) + 3 (- 3)}{4} < 12$

Step 2.8.3

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

$\frac{3 (- x - 3)}{4} < 12$

Step 2.9

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

$\frac{3 (- (x) - 3)}{4} < 12$

Step 2.10

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

$\frac{3 (- (x) - 1 (3))}{4} < 12$

Step 2.11

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

$\frac{3 (- (x + 3))}{4} < 12$

Step 2.12

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

$\frac{3 (- 1 (x + 3))}{4} < 12$

Step 2.13

Move the negative in front of the fraction.

$- \frac{3 (x + 3)}{4} < 12$

Step 3

Divide each term in $- \frac{3 (x + 3)}{4} < 12$ by $- 1$ and simplify.

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Step 3.1

Divide each term in $- \frac{3 (x + 3)}{4} < 12$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{3 (x + 3)}{4}}{- 1} > \frac{12}{- 1}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{3 (x + 3)}{4}}{1} > \frac{12}{- 1}$

Step 3.2.2

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

$\frac{3 (x + 3)}{4} > \frac{12}{- 1}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $12$ by $- 1$ .

$\frac{3 (x + 3)}{4} > - 12$

Step 4

Multiply both sides by $4$ .

$\frac{3 (x + 3)}{4} \cdot 4 > - 12 \cdot 4$

Step 5

Simplify.

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Step 5.1

Simplify the left side.

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Step 5.1.1

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

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Step 5.1.1.1

Cancel the common factor of $4$ .

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Step 5.1.1.1.1

Cancel the common factor.

$\frac{3 (x + 3)}{4} \cdot 4 > - 12 \cdot 4$

Step 5.1.1.1.2

Rewrite the expression.

$3 (x + 3) > - 12 \cdot 4$

Step 5.1.1.2

Apply the distributive property.

$3 x + 3 \cdot 3 > - 12 \cdot 4$

Step 5.1.1.3

Multiply $3$ by $3$ .

$3 x + 9 > - 12 \cdot 4$

Step 5.2

Simplify the right side.

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Step 5.2.1

Multiply $- 12$ by $4$ .

$3 x + 9 > - 48$

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

Subtract $9$ from both sides of the inequality.

$3 x > - 48 - 9$

Step 6.1.2

Subtract $9$ from $- 48$ .

$3 x > - 57$

Step 6.2

Divide each term in $3 x > - 57$ by $3$ and simplify.

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Step 6.2.1

Divide each term in $3 x > - 57$ by $3$ .

$\frac{3 x}{3} > \frac{- 57}{3}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Cancel the common factor of $3$ .

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Step 6.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} > \frac{- 57}{3}$

Step 6.2.2.1.2

Divide $x$ by $1$ .

$x > \frac{- 57}{3}$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Divide $- 57$ by $3$ .

$x > - 19$

Step 7

The result can be shown in multiple forms.

Inequality Form:

$x > - 19$

Interval Notation:

$(- 19, \infty)$