Finite Math Examples

$\frac{m}{2} - 3 > \frac{7}{2} - \frac{m}{5}$

Step 1

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

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Step 1.1

Add $\frac{m}{5}$ to both sides of the inequality.

$\frac{m}{2} - 3 + \frac{m}{5} > \frac{7}{2}$

Step 1.2

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

$\frac{m}{2} \cdot \frac{5}{5} + \frac{m}{5} - 3 > \frac{7}{2}$

Step 1.3

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

$\frac{m}{2} \cdot \frac{5}{5} + \frac{m}{5} \cdot \frac{2}{2} - 3 > \frac{7}{2}$

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 $\frac{m}{2}$ by $\frac{5}{5}$ .

$\frac{m \cdot 5}{2 \cdot 5} + \frac{m}{5} \cdot \frac{2}{2} - 3 > \frac{7}{2}$

Step 1.4.2

Multiply $2$ by $5$ .

$\frac{m \cdot 5}{10} + \frac{m}{5} \cdot \frac{2}{2} - 3 > \frac{7}{2}$

Step 1.4.3

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

$\frac{m \cdot 5}{10} + \frac{m \cdot 2}{5 \cdot 2} - 3 > \frac{7}{2}$

Step 1.4.4

Multiply $5$ by $2$ .

$\frac{m \cdot 5}{10} + \frac{m \cdot 2}{10} - 3 > \frac{7}{2}$

Step 1.5

Combine the numerators over the common denominator.

$\frac{m \cdot 5 + m \cdot 2}{10} - 3 > \frac{7}{2}$

Step 1.6

Reorder $m$ and $2$ .

$\frac{5 \cdot m + 2 \cdot m}{10} - 3 > \frac{7}{2}$

Step 1.7

Add $5 \cdot m$ and $2 \cdot m$ .

$\frac{7 m}{10} - 3 > \frac{7}{2}$

Step 2

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

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Step 2.1

Add $3$ to both sides of the inequality.

$\frac{7 m}{10} > \frac{7}{2} + 3$

Step 2.2

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

$\frac{7 m}{10} > \frac{7}{2} + 3 \cdot \frac{2}{2}$

Step 2.3

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

$\frac{7 m}{10} > \frac{7}{2} + \frac{3 \cdot 2}{2}$

Step 2.4

Combine the numerators over the common denominator.

$\frac{7 m}{10} > \frac{7 + 3 \cdot 2}{2}$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Multiply $3$ by $2$ .

$\frac{7 m}{10} > \frac{7 + 6}{2}$

Step 2.5.2

Add $7$ and $6$ .

$\frac{7 m}{10} > \frac{13}{2}$

Step 3

Multiply both sides by $10$ .

$\frac{7 m}{10} \cdot 10 > \frac{13}{2} \cdot 10$

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 $10$ .

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Step 4.1.1.1

Cancel the common factor.

$\frac{7 m}{10} \cdot 10 > \frac{13}{2} \cdot 10$

Step 4.1.1.2

Rewrite the expression.

$7 m > \frac{13}{2} \cdot 10$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{13}{2} \cdot 10$ .

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Step 4.2.1.1

Cancel the common factor of $2$ .

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Step 4.2.1.1.1

Factor $2$ out of $10$ .

$7 m > \frac{13}{2} \cdot (2 (5))$

Step 4.2.1.1.2

Cancel the common factor.

$7 m > \frac{13}{2} \cdot (2 \cdot 5)$

Step 4.2.1.1.3

Rewrite the expression.

$7 m > 13 \cdot 5$

Step 4.2.1.2

Multiply $13$ by $5$ .

$7 m > 65$

Step 5

Divide each term in $7 m > 65$ by $7$ and simplify.

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Step 5.1

Divide each term in $7 m > 65$ by $7$ .

$\frac{7 m}{7} > \frac{65}{7}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $7$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{7 m}{7} > \frac{65}{7}$

Step 5.2.1.2

Divide $m$ by $1$ .

$m > \frac{65}{7}$

Step 6

The result can be shown in multiple forms.

Inequality Form:

$m > \frac{65}{7}$

Interval Notation:

$(\frac{65}{7}, \infty)$