Finite Math Examples

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

Step 1

Simplify $\frac{1}{4} \cdot (2 x - 1)$ .

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Step 1.1

Rewrite.

$0 + 0 + \frac{1}{4} \cdot (2 x - 1) > \frac{1}{6} \cdot (3 x - 2)$

Step 1.2

Simplify by adding zeros.

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

Step 1.3

Apply the distributive property.

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

Step 1.4

Cancel the common factor of $2$ .

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Step 1.4.1

Factor $2$ out of $4$ .

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

Step 1.4.2

Factor $2$ out of $2 x$ .

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

Step 1.4.3

Cancel the common factor.

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

Step 1.4.4

Rewrite the expression.

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

Step 1.5

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

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

Step 1.6

Combine $\frac{1}{4}$ and $- 1$ .

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

Step 1.7

Move the negative in front of the fraction.

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

Step 2

Simplify $\frac{1}{6} \cdot (3 x - 2)$ .

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Step 2.1

Apply the distributive property.

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

Step 2.2

Cancel the common factor of $3$ .

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Step 2.2.1

Factor $3$ out of $6$ .

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

Step 2.2.2

Factor $3$ out of $3 x$ .

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

Step 2.2.3

Cancel the common factor.

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

Step 2.2.4

Rewrite the expression.

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

Step 2.3

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

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

Step 2.4

Cancel the common factor of $2$ .

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Step 2.4.1

Factor $2$ out of $6$ .

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

Step 2.4.2

Factor $2$ out of $- 2$ .

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

Step 2.4.3

Cancel the common factor.

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

Step 2.4.4

Rewrite the expression.

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

Step 2.5

Combine $\frac{1}{3}$ and $- 1$ .

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

Step 2.6

Move the negative in front of the fraction.

$\frac{x}{2} - \frac{1}{4} > \frac{x}{2} - \frac{1}{3}$

Step 3

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

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Step 3.1

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

$\frac{x}{2} - \frac{1}{4} - \frac{x}{2} > - \frac{1}{3}$

Step 3.2

Combine the opposite terms in $\frac{x}{2} - \frac{1}{4} - \frac{x}{2}$ .

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Step 3.2.1

Subtract $\frac{x}{2}$ from $\frac{x}{2}$ .

$0 - \frac{1}{4} > - \frac{1}{3}$

Step 3.2.2

Subtract $\frac{1}{4}$ from $0$ .

$- \frac{1}{4} > - \frac{1}{3}$

Step 4

Since $- \frac{1}{4} > - \frac{1}{3}$ , the inequality will always be true for any value of $x$ .

All real numbers

Step 5

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$