Calculus Examples

$\frac{u}{2} - \frac{2}{3} < \frac{u}{3} + 2$

Step 1

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

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Step 1.1

Subtract $\frac{u}{3}$ from both sides of the inequality.

$\frac{u}{2} - \frac{2}{3} - \frac{u}{3} < 2$

Step 1.2

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

$\frac{u}{2} \cdot \frac{3}{3} - \frac{u}{3} - \frac{2}{3} < 2$

Step 1.3

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

$\frac{u}{2} \cdot \frac{3}{3} - \frac{u}{3} \cdot \frac{2}{2} - \frac{2}{3} < 2$

Step 1.4

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

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Step 1.4.1

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

$\frac{u \cdot 3}{2 \cdot 3} - \frac{u}{3} \cdot \frac{2}{2} - \frac{2}{3} < 2$

Step 1.4.2

Multiply $2$ by $3$ .

$\frac{u \cdot 3}{6} - \frac{u}{3} \cdot \frac{2}{2} - \frac{2}{3} < 2$

Step 1.4.3

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

$\frac{u \cdot 3}{6} - \frac{u \cdot 2}{3 \cdot 2} - \frac{2}{3} < 2$

Step 1.4.4

Multiply $3$ by $2$ .

$\frac{u \cdot 3}{6} - \frac{u \cdot 2}{6} - \frac{2}{3} < 2$

Step 1.5

Combine the numerators over the common denominator.

$\frac{u \cdot 3 - u \cdot 2}{6} - \frac{2}{3} < 2$

Step 1.6

Simplify each term.

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Step 1.6.1

Simplify the numerator.

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Step 1.6.1.1

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

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Step 1.6.1.1.1

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

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

Step 1.6.1.1.2

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

$\frac{u (3) + u (- 1 \cdot 2)}{6} - \frac{2}{3} < 2$

Step 1.6.1.1.3

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

$\frac{u (3 - 1 \cdot 2)}{6} - \frac{2}{3} < 2$

Step 1.6.1.2

Multiply $- 1$ by $2$ .

$\frac{u (3 - 2)}{6} - \frac{2}{3} < 2$

Step 1.6.1.3

Subtract $2$ from $3$ .

$\frac{u \cdot 1}{6} - \frac{2}{3} < 2$

Step 1.6.2

Multiply $u$ by $1$ .

$\frac{u}{6} - \frac{2}{3} < 2$

Step 2

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

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Step 2.1

Add $\frac{2}{3}$ to both sides of the inequality.

$\frac{u}{6} < 2 + \frac{2}{3}$

Step 2.2

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

$\frac{u}{6} < 2 \cdot \frac{3}{3} + \frac{2}{3}$

Step 2.3

Combine $2$ and $\frac{3}{3}$ .

$\frac{u}{6} < \frac{2 \cdot 3}{3} + \frac{2}{3}$

Step 2.4

Combine the numerators over the common denominator.

$\frac{u}{6} < \frac{2 \cdot 3 + 2}{3}$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Multiply $2$ by $3$ .

$\frac{u}{6} < \frac{6 + 2}{3}$

Step 2.5.2

Add $6$ and $2$ .

$\frac{u}{6} < \frac{8}{3}$

Step 3

Multiply both sides by $6$ .

$\frac{u}{6} \cdot 6 < \frac{8}{3} \cdot 6$

Step 4

Simplify.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Cancel the common factor of $6$ .

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Step 4.1.1.1

Cancel the common factor.

$\frac{u}{6} \cdot 6 < \frac{8}{3} \cdot 6$

Step 4.1.1.2

Rewrite the expression.

$u < \frac{8}{3} \cdot 6$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{8}{3} \cdot 6$ .

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Step 4.2.1.1

Cancel the common factor of $3$ .

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Step 4.2.1.1.1

Factor $3$ out of $6$ .

$u < \frac{8}{3} \cdot (3 (2))$

Step 4.2.1.1.2

Cancel the common factor.

$u < \frac{8}{3} \cdot (3 \cdot 2)$

Step 4.2.1.1.3

Rewrite the expression.

$u < 8 \cdot 2$

Step 4.2.1.2

Multiply $8$ by $2$ .

$u < 16$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$u < 16$

Interval Notation:

$(- \infty, 16)$