Finite Math Examples

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

Step 1

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

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Step 1.1

Subtract $\frac{5 x}{2}$ from both sides of the inequality.

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

Step 1.2

To write $\frac{2 x}{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 1.3

To write $- \frac{5 x}{2}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

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

Step 1.4

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.4.1

Multiply $\frac{2 x}{5}$ by $\frac{2}{2}$ .

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

Step 1.4.2

Multiply $5$ by $2$ .

$\frac{2 x \cdot 2}{10} - \frac{5 x}{2} \cdot \frac{5}{5} - 1 > 2$

Step 1.4.3

Multiply $\frac{5 x}{2}$ by $\frac{5}{5}$ .

$\frac{2 x \cdot 2}{10} - \frac{5 x \cdot 5}{2 \cdot 5} - 1 > 2$

Step 1.4.4

Multiply $2$ by $5$ .

$\frac{2 x \cdot 2}{10} - \frac{5 x \cdot 5}{10} - 1 > 2$

Step 1.5

Combine the numerators over the common denominator.

$\frac{2 x \cdot 2 - 5 x \cdot 5}{10} - 1 > 2$

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 $x$ out of $2 x \cdot 2 - 5 x \cdot 5$ .

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Step 1.6.1.1.1

Factor $x$ out of $2 x \cdot 2$ .

$\frac{x (2 \cdot 2) - 5 x \cdot 5}{10} - 1 > 2$

Step 1.6.1.1.2

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

$\frac{x (2 \cdot 2) + x (- 5 \cdot 5)}{10} - 1 > 2$

Step 1.6.1.1.3

Factor $x$ out of $x (2 \cdot 2) + x (- 5 \cdot 5)$ .

$\frac{x (2 \cdot 2 - 5 \cdot 5)}{10} - 1 > 2$

Step 1.6.1.2

Multiply $2$ by $2$ .

$\frac{x (4 - 5 \cdot 5)}{10} - 1 > 2$

Step 1.6.1.3

Multiply $- 5$ by $5$ .

$\frac{x (4 - 25)}{10} - 1 > 2$

Step 1.6.1.4

Subtract $25$ from $4$ .

$\frac{x \cdot - 21}{10} - 1 > 2$

Step 1.6.2

Move $- 21$ to the left of $x$ .

$\frac{- 21 \cdot x}{10} - 1 > 2$

Step 1.6.3

Move the negative in front of the fraction.

$- \frac{21 x}{10} - 1 > 2$

Step 2

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

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Step 2.1

Add $1$ to both sides of the inequality.

$- \frac{21 x}{10} > 2 + 1$

Step 2.2

Add $2$ and $1$ .

$- \frac{21 x}{10} > 3$

Step 3

Divide each term in $- \frac{21 x}{10} > 3$ by $- 1$ and simplify.

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Step 3.1

Divide each term in $- \frac{21 x}{10} > 3$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{21 x}{10}}{- 1} < \frac{3}{- 1}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{21 x}{10}}{1} < \frac{3}{- 1}$

Step 3.2.2

Divide $\frac{21 x}{10}$ by $1$ .

$\frac{21 x}{10} < \frac{3}{- 1}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $3$ by $- 1$ .

$\frac{21 x}{10} < - 3$

Step 4

Multiply both sides by $10$ .

$\frac{21 x}{10} \cdot 10 < - 3 \cdot 10$

Step 5

Simplify.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Cancel the common factor of $10$ .

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Step 5.1.1.1

Cancel the common factor.

$\frac{21 x}{10} \cdot 10 < - 3 \cdot 10$

Step 5.1.1.2

Rewrite the expression.

$21 x < - 3 \cdot 10$

Step 5.2

Simplify the right side.

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Step 5.2.1

Multiply $- 3$ by $10$ .

$21 x < - 30$

Step 6

Divide each term in $21 x < - 30$ by $21$ and simplify.

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Step 6.1

Divide each term in $21 x < - 30$ by $21$ .

$\frac{21 x}{21} < \frac{- 30}{21}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $21$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{21 x}{21} < \frac{- 30}{21}$

Step 6.2.1.2

Divide $x$ by $1$ .

$x < \frac{- 30}{21}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Cancel the common factor of $- 30$ and $21$ .

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Step 6.3.1.1

Factor $3$ out of $- 30$ .

$x < \frac{3 (- 10)}{21}$

Step 6.3.1.2

Cancel the common factors.

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Step 6.3.1.2.1

Factor $3$ out of $21$ .

$x < \frac{3 \cdot - 10}{3 \cdot 7}$

Step 6.3.1.2.2

Cancel the common factor.

$x < \frac{3 \cdot - 10}{3 \cdot 7}$

Step 6.3.1.2.3

Rewrite the expression.

$x < \frac{- 10}{7}$

Step 6.3.2

Move the negative in front of the fraction.

$x < - \frac{10}{7}$

Step 7

The result can be shown in multiple forms.

Inequality Form:

$x < - \frac{10}{7}$

Interval Notation:

$(- \infty, - \frac{10}{7})$