Algebra Examples

$\frac{x - 6}{4} < \frac{x - 3}{6} + \frac{1}{12}$

Step 1

Multiply both sides by $4$ .

$\frac{x - 6}{4} \cdot 4 < (\frac{x - 3}{6} + \frac{1}{12}) \cdot 4$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Cancel the common factor of $4$ .

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Step 2.1.1.1

Cancel the common factor.

$\frac{x - 6}{4} \cdot 4 < (\frac{x - 3}{6} + \frac{1}{12}) \cdot 4$

Step 2.1.1.2

Rewrite the expression.

$x - 6 < (\frac{x - 3}{6} + \frac{1}{12}) \cdot 4$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $(\frac{x - 3}{6} + \frac{1}{12}) \cdot 4$ .

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Step 2.2.1.1

To write $\frac{x - 3}{6}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x - 6 < (\frac{x - 3}{6} \cdot \frac{2}{2} + \frac{1}{12}) \cdot 4$

Step 2.2.1.2

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.2.1.2.1

Multiply $\frac{x - 3}{6}$ by $\frac{2}{2}$ .

$x - 6 < (\frac{(x - 3) \cdot 2}{6 \cdot 2} + \frac{1}{12}) \cdot 4$

Step 2.2.1.2.2

Multiply $6$ by $2$ .

$x - 6 < (\frac{(x - 3) \cdot 2}{12} + \frac{1}{12}) \cdot 4$

Step 2.2.1.3

Combine the numerators over the common denominator.

$x - 6 < \frac{(x - 3) \cdot 2 + 1}{12} \cdot 4$

Step 2.2.1.4

Simplify the numerator.

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Step 2.2.1.4.1

Apply the distributive property.

$x - 6 < \frac{x \cdot 2 - 3 \cdot 2 + 1}{12} \cdot 4$

Step 2.2.1.4.2

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

$x - 6 < \frac{2 \cdot x - 3 \cdot 2 + 1}{12} \cdot 4$

Step 2.2.1.4.3

Multiply $- 3$ by $2$ .

$x - 6 < \frac{2 \cdot x - 6 + 1}{12} \cdot 4$

Step 2.2.1.4.4

Add $- 6$ and $1$ .

$x - 6 < \frac{2 x - 5}{12} \cdot 4$

Step 2.2.1.5

Cancel the common factor of $4$ .

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Step 2.2.1.5.1

Factor $4$ out of $12$ .

$x - 6 < \frac{2 x - 5}{4 (3)} \cdot 4$

Step 2.2.1.5.2

Cancel the common factor.

$x - 6 < \frac{2 x - 5}{4 \cdot 3} \cdot 4$

Step 2.2.1.5.3

Rewrite the expression.

$x - 6 < \frac{2 x - 5}{3}$

Step 3

Solve for $x$ .

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Step 3.1

Multiply both sides by $3$ .

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

Step 3.2

Simplify.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $(x - 6) \cdot 3$ .

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Step 3.2.1.1.1

Apply the distributive property.

$x \cdot 3 - 6 \cdot 3 < \frac{2 x - 5}{3} \cdot 3$

Step 3.2.1.1.2

Simplify the expression.

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Step 3.2.1.1.2.1

Move $3$ to the left of $x$ .

$3 \cdot x - 6 \cdot 3 < \frac{2 x - 5}{3} \cdot 3$

Step 3.2.1.1.2.2

Multiply $- 6$ by $3$ .

$3 x - 18 < \frac{2 x - 5}{3} \cdot 3$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Cancel the common factor of $3$ .

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Step 3.2.2.1.1

Cancel the common factor.

$3 x - 18 < \frac{2 x - 5}{3} \cdot 3$

Step 3.2.2.1.2

Rewrite the expression.

$3 x - 18 < 2 x - 5$

Step 3.3

Solve for $x$ .

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Step 3.3.1

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

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Step 3.3.1.1

Subtract $2 x$ from both sides of the inequality.

$3 x - 18 - 2 x < - 5$

Step 3.3.1.2

Subtract $2 x$ from $3 x$ .

$x - 18 < - 5$

Step 3.3.2

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

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Step 3.3.2.1

Add $18$ to both sides of the inequality.

$x < - 5 + 18$

Step 3.3.2.2

Add $- 5$ and $18$ .

$x < 13$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$x < 13$

Interval Notation:

$(- \infty, 13)$