Finite Math Examples

$3 x - \frac{1}{2} > \frac{10}{3} x + 2$

Step 1

Combine $\frac{10}{3}$ and $x$ .

$3 x - \frac{1}{2} > \frac{10 x}{3} + 2$

Step 2

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

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Step 2.1

Subtract $\frac{10 x}{3}$ from both sides of the inequality.

$3 x - \frac{1}{2} - \frac{10 x}{3} > 2$

Step 2.2

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

$3 x \cdot \frac{3}{3} - \frac{10 x}{3} - \frac{1}{2} > 2$

Step 2.3

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

$\frac{3 x \cdot 3}{3} - \frac{10 x}{3} - \frac{1}{2} > 2$

Step 2.4

Combine the numerators over the common denominator.

$\frac{3 x \cdot 3 - 10 x}{3} - \frac{1}{2} > 2$

Step 2.5

Simplify each term.

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Step 2.5.1

Simplify the numerator.

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Step 2.5.1.1

Factor $x$ out of $3 x \cdot 3 - 10 x$ .

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Step 2.5.1.1.1

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

$\frac{x (3 \cdot 3) - 10 x}{3} - \frac{1}{2} > 2$

Step 2.5.1.1.2

Factor $x$ out of $- 10 x$ .

$\frac{x (3 \cdot 3) + x \cdot - 10}{3} - \frac{1}{2} > 2$

Step 2.5.1.1.3

Factor $x$ out of $x (3 \cdot 3) + x \cdot - 10$ .

$\frac{x (3 \cdot 3 - 10)}{3} - \frac{1}{2} > 2$

Step 2.5.1.2

Multiply $3$ by $3$ .

$\frac{x (9 - 10)}{3} - \frac{1}{2} > 2$

Step 2.5.1.3

Subtract $10$ from $9$ .

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

Step 2.5.2

Move $- 1$ to the left of $x$ .

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

Step 2.5.3

Move the negative in front of the fraction.

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

Step 3

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

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Step 3.1

Add $\frac{1}{2}$ to both sides of the inequality.

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Combine the numerators over the common denominator.

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

Step 3.5

Simplify the numerator.

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Step 3.5.1

Multiply $2$ by $2$ .

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

Step 3.5.2

Add $4$ and $1$ .

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

Step 4

Divide each term in $- \frac{x}{3} > \frac{5}{2}$ by $- 1$ and simplify.

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Step 4.1

Divide each term in $- \frac{x}{3} > \frac{5}{2}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

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

Step 4.2

Simplify the left side.

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Step 4.2.1

Dividing two negative values results in a positive value.

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

Step 4.2.2

Divide $\frac{x}{3}$ by $1$ .

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

Step 4.3

Simplify the right side.

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Step 4.3.1

Move the negative one from the denominator of $\frac{\frac{5}{2}}{- 1}$ .

$\frac{x}{3} < - 1 \cdot \frac{5}{2}$

Step 4.3.2

Rewrite $- 1 \cdot \frac{5}{2}$ as $- \frac{5}{2}$ .

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

Step 5

Multiply both sides by $3$ .

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

Step 6

Simplify.

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Step 6.1

Simplify the left side.

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Step 6.1.1

Cancel the common factor of $3$ .

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Step 6.1.1.1

Cancel the common factor.

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

Step 6.1.1.2

Rewrite the expression.

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

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $- \frac{5}{2} \cdot 3$ .

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Step 6.2.1.1

Multiply $- \frac{5}{2} \cdot 3$ .

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Step 6.2.1.1.1

Multiply $3$ by $- 1$ .

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

Step 6.2.1.1.2

Combine $- 3$ and $\frac{5}{2}$ .

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

Step 6.2.1.1.3

Multiply $- 3$ by $5$ .

$x < \frac{- 15}{2}$

Step 6.2.1.2

Move the negative in front of the fraction.

$x < - \frac{15}{2}$

Step 7

Convert the inequality to interval notation.

$(- \infty, - \frac{15}{2})$

Step 8