Precalculus Examples

$\frac{1}{2} x - \frac{2}{3} > 3$

Step 1

Combine $\frac{1}{2}$ and $x$ .

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

Step 2

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

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Step 2.1

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

$\frac{x}{2} > 3 + \frac{2}{3}$

Step 2.2

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

$\frac{x}{2} > 3 \cdot \frac{3}{3} + \frac{2}{3}$

Step 2.3

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

$\frac{x}{2} > \frac{3 \cdot 3}{3} + \frac{2}{3}$

Step 2.4

Combine the numerators over the common denominator.

$\frac{x}{2} > \frac{3 \cdot 3 + 2}{3}$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Multiply $3$ by $3$ .

$\frac{x}{2} > \frac{9 + 2}{3}$

Step 2.5.2

Add $9$ and $2$ .

$\frac{x}{2} > \frac{11}{3}$

Step 3

Multiply both sides by $2$ .

$\frac{x}{2} \cdot 2 > \frac{11}{3} \cdot 2$

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 $2$ .

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Step 4.1.1.1

Cancel the common factor.

$\frac{x}{2} \cdot 2 > \frac{11}{3} \cdot 2$

Step 4.1.1.2

Rewrite the expression.

$x > \frac{11}{3} \cdot 2$

Step 4.2

Simplify the right side.

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Step 4.2.1

Multiply $\frac{11}{3} \cdot 2$ .

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Step 4.2.1.1

Combine $\frac{11}{3}$ and $2$ .

$x > \frac{11 \cdot 2}{3}$

Step 4.2.1.2

Multiply $11$ by $2$ .

$x > \frac{22}{3}$

Step 5

Convert the inequality to interval notation.

$(\frac{22}{3}, \infty)$

Step 6