Calculus Examples

$\frac{x}{2} - \frac{x}{3} > 5$

Step 1

Simplify $\frac{x}{2} - \frac{x}{3}$ .

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.3.1

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

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

Step 1.3.2

Multiply $2$ by $3$ .

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

Step 1.3.3

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

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

Step 1.3.4

Multiply $3$ by $2$ .

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

Step 1.4

Combine the numerators over the common denominator.

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

Step 1.5

Simplify the numerator.

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Step 1.5.1

Factor $x$ out of $x \cdot 3 - x \cdot 2$ .

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Step 1.5.1.1

Factor $x$ out of $x \cdot 3$ .

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

Step 1.5.1.2

Factor $x$ out of $- x \cdot 2$ .

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

Step 1.5.1.3

Factor $x$ out of $x (3) + x (- 1 \cdot 2)$ .

$\frac{x (3 - 1 \cdot 2)}{6} > 5$

Step 1.5.2

Multiply $- 1$ by $2$ .

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

Step 1.5.3

Subtract $2$ from $3$ .

$\frac{x \cdot 1}{6} > 5$

Step 1.6

Multiply $x$ by $1$ .

$\frac{x}{6} > 5$

Step 2

Multiply both sides by $6$ .

$\frac{x}{6} \cdot 6 > 5 \cdot 6$

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Cancel the common factor of $6$ .

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Step 3.1.1.1

Cancel the common factor.

$\frac{x}{6} \cdot 6 > 5 \cdot 6$

Step 3.1.1.2

Rewrite the expression.

$x > 5 \cdot 6$

Step 3.2

Simplify the right side.

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Step 3.2.1

Multiply $5$ by $6$ .

$x > 30$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$x > 30$

Interval Notation:

$(30, \infty)$