Finite Math Examples

$- 4 - \frac{m}{4} - 3 < \frac{7}{4} - \frac{m}{5} - 4$

Step 1

Subtract $3$ from $- 4$ .

$- \frac{m}{4} - 7 < \frac{7}{4} - \frac{m}{5} - 4$

Step 2

Simplify $\frac{7}{4} - \frac{m}{5} - 4$ .

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Step 2.1

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$- \frac{m}{4} - 7 < - \frac{m}{5} + \frac{7}{4} - 4 \cdot \frac{4}{4}$

Step 2.2

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

$- \frac{m}{4} - 7 < - \frac{m}{5} + \frac{7}{4} + \frac{- 4 \cdot 4}{4}$

Step 2.3

Combine the numerators over the common denominator.

$- \frac{m}{4} - 7 < - \frac{m}{5} + \frac{7 - 4 \cdot 4}{4}$

Step 2.4

Simplify the numerator.

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Step 2.4.1

Multiply $- 4$ by $4$ .

$- \frac{m}{4} - 7 < - \frac{m}{5} + \frac{7 - 16}{4}$

Step 2.4.2

Subtract $16$ from $7$ .

$- \frac{m}{4} - 7 < - \frac{m}{5} + \frac{- 9}{4}$

Step 2.5

Move the negative in front of the fraction.

$- \frac{m}{4} - 7 < - \frac{m}{5} - \frac{9}{4}$

Step 3

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

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Step 3.1

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

$- \frac{m}{4} - 7 + \frac{m}{5} < - \frac{9}{4}$

Step 3.2

To write $- \frac{m}{4}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$- \frac{m}{4} \cdot \frac{5}{5} + \frac{m}{5} - 7 < - \frac{9}{4}$

Step 3.3

To write $\frac{m}{5}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$- \frac{m}{4} \cdot \frac{5}{5} + \frac{m}{5} \cdot \frac{4}{4} - 7 < - \frac{9}{4}$

Step 3.4

Write each expression with a common denominator of $20$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.1

Multiply $\frac{m}{4}$ by $\frac{5}{5}$ .

$- \frac{m \cdot 5}{4 \cdot 5} + \frac{m}{5} \cdot \frac{4}{4} - 7 < - \frac{9}{4}$

Step 3.4.2

Multiply $4$ by $5$ .

$- \frac{m \cdot 5}{20} + \frac{m}{5} \cdot \frac{4}{4} - 7 < - \frac{9}{4}$

Step 3.4.3

Multiply $\frac{m}{5}$ by $\frac{4}{4}$ .

$- \frac{m \cdot 5}{20} + \frac{m \cdot 4}{5 \cdot 4} - 7 < - \frac{9}{4}$

Step 3.4.4

Multiply $5$ by $4$ .

$- \frac{m \cdot 5}{20} + \frac{m \cdot 4}{20} - 7 < - \frac{9}{4}$

Step 3.5

Combine the numerators over the common denominator.

$\frac{- m \cdot 5 + m \cdot 4}{20} - 7 < - \frac{9}{4}$

Step 3.6

Simplify each term.

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Step 3.6.1

Simplify the numerator.

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Step 3.6.1.1

Factor $m$ out of $- m \cdot 5 + m \cdot 4$ .

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Step 3.6.1.1.1

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

$\frac{m (- 1 \cdot 5) + m \cdot 4}{20} - 7 < - \frac{9}{4}$

Step 3.6.1.1.2

Factor $m$ out of $m \cdot 4$ .

$\frac{m (- 1 \cdot 5) + m (4)}{20} - 7 < - \frac{9}{4}$

Step 3.6.1.1.3

Factor $m$ out of $m (- 1 \cdot 5) + m (4)$ .

$\frac{m (- 1 \cdot 5 + 4)}{20} - 7 < - \frac{9}{4}$

Step 3.6.1.2

Multiply $- 1$ by $5$ .

$\frac{m (- 5 + 4)}{20} - 7 < - \frac{9}{4}$

Step 3.6.1.3

Add $- 5$ and $4$ .

$\frac{m \cdot - 1}{20} - 7 < - \frac{9}{4}$

Step 3.6.2

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

$\frac{- 1 \cdot m}{20} - 7 < - \frac{9}{4}$

Step 3.6.3

Move the negative in front of the fraction.

$- \frac{m}{20} - 7 < - \frac{9}{4}$

Step 4

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

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Step 4.1

Add $7$ to both sides of the inequality.

$- \frac{m}{20} < - \frac{9}{4} + 7$

Step 4.2

To write $7$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$- \frac{m}{20} < - \frac{9}{4} + 7 \cdot \frac{4}{4}$

Step 4.3

Combine $7$ and $\frac{4}{4}$ .

$- \frac{m}{20} < - \frac{9}{4} + \frac{7 \cdot 4}{4}$

Step 4.4

Combine the numerators over the common denominator.

$- \frac{m}{20} < \frac{- 9 + 7 \cdot 4}{4}$

Step 4.5

Simplify the numerator.

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Step 4.5.1

Multiply $7$ by $4$ .

$- \frac{m}{20} < \frac{- 9 + 28}{4}$

Step 4.5.2

Add $- 9$ and $28$ .

$- \frac{m}{20} < \frac{19}{4}$

Step 5

Divide each term in $- \frac{m}{20} < \frac{19}{4}$ by $- 1$ and simplify.

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Step 5.1

Divide each term in $- \frac{m}{20} < \frac{19}{4}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{m}{20}}{- 1} > \frac{\frac{19}{4}}{- 1}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{m}{20}}{1} > \frac{\frac{19}{4}}{- 1}$

Step 5.2.2

Divide $\frac{m}{20}$ by $1$ .

$\frac{m}{20} > \frac{\frac{19}{4}}{- 1}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Move the negative one from the denominator of $\frac{\frac{19}{4}}{- 1}$ .

$\frac{m}{20} > - 1 \cdot \frac{19}{4}$

Step 5.3.2

Rewrite $- 1 \cdot \frac{19}{4}$ as $- \frac{19}{4}$ .

$\frac{m}{20} > - \frac{19}{4}$

Step 6

Multiply both sides by $20$ .

$\frac{m}{20} \cdot 20 > - \frac{19}{4} \cdot 20$

Step 7

Simplify.

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Step 7.1

Simplify the left side.

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Step 7.1.1

Cancel the common factor of $20$ .

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Step 7.1.1.1

Cancel the common factor.

$\frac{m}{20} \cdot 20 > - \frac{19}{4} \cdot 20$

Step 7.1.1.2

Rewrite the expression.

$m > - \frac{19}{4} \cdot 20$

Step 7.2

Simplify the right side.

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Step 7.2.1

Simplify $- \frac{19}{4} \cdot 20$ .

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Step 7.2.1.1

Cancel the common factor of $4$ .

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Step 7.2.1.1.1

Move the leading negative in $- \frac{19}{4}$ into the numerator.

$m > \frac{- 19}{4} \cdot 20$

Step 7.2.1.1.2

Factor $4$ out of $20$ .

$m > \frac{- 19}{4} \cdot (4 (5))$

Step 7.2.1.1.3

Cancel the common factor.

$m > \frac{- 19}{4} \cdot (4 \cdot 5)$

Step 7.2.1.1.4

Rewrite the expression.

$m > - 19 \cdot 5$

Step 7.2.1.2

Multiply $- 19$ by $5$ .

$m > - 95$

Step 8

The result can be shown in multiple forms.

Inequality Form:

$m > - 95$

Interval Notation:

$(- 95, \infty)$