Calculus Examples

$3 x - \frac{5}{2} > \frac{1}{2} x$

Step 1

Simplify $\frac{1}{2} x$ .

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Step 1.1

Rewrite.

$3 x - \frac{5}{2} > 0 + 0 + \frac{1}{2} x$

Step 1.2

Simplify by adding zeros.

$3 x - \frac{5}{2} > \frac{1}{2} x$

Step 1.3

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

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

Step 2

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

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Step 2.1

Subtract $\frac{x}{2}$ from both sides of the inequality.

$3 x - \frac{5}{2} - \frac{x}{2} > 0$

Step 2.2

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

$3 x \cdot \frac{2}{2} - \frac{x}{2} - \frac{5}{2} > 0$

Step 2.3

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

$\frac{3 x \cdot 2}{2} - \frac{x}{2} - \frac{5}{2} > 0$

Step 2.4

Combine the numerators over the common denominator.

$\frac{3 x \cdot 2 - x}{2} - \frac{5}{2} > 0$

Step 2.5

Combine the numerators over the common denominator.

$\frac{3 x \cdot 2 - x - 5}{2} > 0$

Step 2.6

Multiply $2$ by $3$ .

$\frac{6 x - x - 5}{2} > 0$

Step 2.7

Subtract $x$ from $6 x$ .

$\frac{5 x - 5}{2} > 0$

Step 2.8

Factor $5$ out of $5 x - 5$ .

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Step 2.8.1

Factor $5$ out of $5 x$ .

$\frac{5 (x) - 5}{2} > 0$

Step 2.8.2

Factor $5$ out of $- 5$ .

$\frac{5 (x) + 5 (- 1)}{2} > 0$

Step 2.8.3

Factor $5$ out of $5 (x) + 5 (- 1)$ .

$\frac{5 (x - 1)}{2} > 0$

Step 3

Multiply both sides by $2$ .

$\frac{5 (x - 1)}{2} \cdot 2 > 0 \cdot 2$

Step 4

Simplify.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Simplify $\frac{5 (x - 1)}{2} \cdot 2$ .

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Step 4.1.1.1

Cancel the common factor of $2$ .

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Step 4.1.1.1.1

Cancel the common factor.

$\frac{5 (x - 1)}{2} \cdot 2 > 0 \cdot 2$

Step 4.1.1.1.2

Rewrite the expression.

$5 (x - 1) > 0 \cdot 2$

Step 4.1.1.2

Apply the distributive property.

$5 x + 5 \cdot - 1 > 0 \cdot 2$

Step 4.1.1.3

Multiply $5$ by $- 1$ .

$5 x - 5 > 0 \cdot 2$

Step 4.2

Simplify the right side.

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Step 4.2.1

Multiply $0$ by $2$ .

$5 x - 5 > 0$

Step 5

Solve for $x$ .

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Step 5.1

Add $5$ to both sides of the inequality.

$5 x > 5$

Step 5.2

Divide each term in $5 x > 5$ by $5$ and simplify.

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Step 5.2.1

Divide each term in $5 x > 5$ by $5$ .

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

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $5$ .

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Step 5.2.2.1.1

Cancel the common factor.

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

Step 5.2.2.1.2

Divide $x$ by $1$ .

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

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Divide $5$ by $5$ .

$x > 1$

Step 6

The result can be shown in multiple forms.

Inequality Form:

$x > 1$

Interval Notation:

$(1, \infty)$