Algebra Examples

$\frac{x - 4}{6} \geq \frac{x - 2}{9} + \frac{5}{18}$

Step 1

Multiply both sides by $6$ .

$\frac{x - 4}{6} \cdot 6 \geq (\frac{x - 2}{9} + \frac{5}{18}) \cdot 6$

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 $6$ .

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Step 2.1.1.1

Cancel the common factor.

$\frac{x - 4}{6} \cdot 6 \geq (\frac{x - 2}{9} + \frac{5}{18}) \cdot 6$

Step 2.1.1.2

Rewrite the expression.

$x - 4 \geq (\frac{x - 2}{9} + \frac{5}{18}) \cdot 6$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $(\frac{x - 2}{9} + \frac{5}{18}) \cdot 6$ .

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Step 2.2.1.1

To write $\frac{x - 2}{9}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x - 4 \geq (\frac{x - 2}{9} \cdot \frac{2}{2} + \frac{5}{18}) \cdot 6$

Step 2.2.1.2

Write each expression with a common denominator of $18$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.2.1.2.1

Multiply $\frac{x - 2}{9}$ by $\frac{2}{2}$ .

$x - 4 \geq (\frac{(x - 2) \cdot 2}{9 \cdot 2} + \frac{5}{18}) \cdot 6$

Step 2.2.1.2.2

Multiply $9$ by $2$ .

$x - 4 \geq (\frac{(x - 2) \cdot 2}{18} + \frac{5}{18}) \cdot 6$

Step 2.2.1.3

Combine the numerators over the common denominator.

$x - 4 \geq \frac{(x - 2) \cdot 2 + 5}{18} \cdot 6$

Step 2.2.1.4

Simplify the numerator.

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Step 2.2.1.4.1

Apply the distributive property.

$x - 4 \geq \frac{x \cdot 2 - 2 \cdot 2 + 5}{18} \cdot 6$

Step 2.2.1.4.2

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

$x - 4 \geq \frac{2 \cdot x - 2 \cdot 2 + 5}{18} \cdot 6$

Step 2.2.1.4.3

Multiply $- 2$ by $2$ .

$x - 4 \geq \frac{2 \cdot x - 4 + 5}{18} \cdot 6$

Step 2.2.1.4.4

Add $- 4$ and $5$ .

$x - 4 \geq \frac{2 x + 1}{18} \cdot 6$

Step 2.2.1.5

Cancel the common factor of $6$ .

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Step 2.2.1.5.1

Factor $6$ out of $18$ .

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

Step 2.2.1.5.2

Cancel the common factor.

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

Step 2.2.1.5.3

Rewrite the expression.

$x - 4 \geq \frac{2 x + 1}{3}$

Step 3

Solve for $x$ .

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Step 3.1

Multiply both sides by $3$ .

$(x - 4) \cdot 3 \geq \frac{2 x + 1}{3} \cdot 3$

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 $(x - 4) \cdot 3$ .

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Step 3.2.1.1.1

Apply the distributive property.

$x \cdot 3 - 4 \cdot 3 \geq \frac{2 x + 1}{3} \cdot 3$

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 $x$ .

$3 \cdot x - 4 \cdot 3 \geq \frac{2 x + 1}{3} \cdot 3$

Step 3.2.1.1.2.2

Multiply $- 4$ by $3$ .

$3 x - 12 \geq \frac{2 x + 1}{3} \cdot 3$

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.

$3 x - 12 \geq \frac{2 x + 1}{3} \cdot 3$

Step 3.2.2.1.2

Rewrite the expression.

$3 x - 12 \geq 2 x + 1$

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.

$3 x - 12 - 2 x \geq 1$

Step 3.3.1.2

Subtract $2 x$ from $3 x$ .

$x - 12 \geq 1$

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 $12$ to both sides of the inequality.

$x \geq 1 + 12$

Step 3.3.2.2

Add $1$ and $12$ .

$x \geq 13$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$x \geq 13$

Interval Notation:

$[13, \infty)$