Finite Math Examples

$\frac{7}{12} x - \frac{7}{4} < \frac{5}{6} x + \frac{3}{4}$

Step 1

Combine $\frac{7}{12}$ and $x$ .

$\frac{7 x}{12} - \frac{7}{4} < \frac{5}{6} x + \frac{3}{4}$

Step 2

Combine $\frac{5}{6}$ and $x$ .

$\frac{7 x}{12} - \frac{7}{4} < \frac{5 x}{6} + \frac{3}{4}$

Step 3

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

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Step 3.1

Subtract $\frac{5 x}{6}$ from both sides of the inequality.

$\frac{7 x}{12} - \frac{7}{4} - \frac{5 x}{6} < \frac{3}{4}$

Step 3.2

To write $- \frac{5 x}{6}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{7 x}{12} - \frac{5 x}{6} \cdot \frac{2}{2} - \frac{7}{4} < \frac{3}{4}$

Step 3.3

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.3.1

Multiply $\frac{5 x}{6}$ by $\frac{2}{2}$ .

$\frac{7 x}{12} - \frac{5 x \cdot 2}{6 \cdot 2} - \frac{7}{4} < \frac{3}{4}$

Step 3.3.2

Multiply $6$ by $2$ .

$\frac{7 x}{12} - \frac{5 x \cdot 2}{12} - \frac{7}{4} < \frac{3}{4}$

Step 3.4

Combine the numerators over the common denominator.

$\frac{7 x - 5 x \cdot 2}{12} - \frac{7}{4} < \frac{3}{4}$

Step 3.5

Simplify each term.

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Step 3.5.1

Simplify the numerator.

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Step 3.5.1.1

Factor $x$ out of $7 x - 5 x \cdot 2$ .

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Step 3.5.1.1.1

Factor $x$ out of $7 x$ .

$\frac{x \cdot 7 - 5 x \cdot 2}{12} - \frac{7}{4} < \frac{3}{4}$

Step 3.5.1.1.2

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

$\frac{x \cdot 7 + x (- 5 \cdot 2)}{12} - \frac{7}{4} < \frac{3}{4}$

Step 3.5.1.1.3

Factor $x$ out of $x \cdot 7 + x (- 5 \cdot 2)$ .

$\frac{x (7 - 5 \cdot 2)}{12} - \frac{7}{4} < \frac{3}{4}$

Step 3.5.1.2

Multiply $- 5$ by $2$ .

$\frac{x (7 - 10)}{12} - \frac{7}{4} < \frac{3}{4}$

Step 3.5.1.3

Subtract $10$ from $7$ .

$\frac{x \cdot - 3}{12} - \frac{7}{4} < \frac{3}{4}$

Step 3.5.2

Cancel the common factor of $- 3$ and $12$ .

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Step 3.5.2.1

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

$\frac{3 (x \cdot - 1)}{12} - \frac{7}{4} < \frac{3}{4}$

Step 3.5.2.2

Cancel the common factors.

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Step 3.5.2.2.1

Factor $3$ out of $12$ .

$\frac{3 (x \cdot - 1)}{3 (4)} - \frac{7}{4} < \frac{3}{4}$

Step 3.5.2.2.2

Cancel the common factor.

$\frac{3 (x \cdot - 1)}{3 \cdot 4} - \frac{7}{4} < \frac{3}{4}$

Step 3.5.2.2.3

Rewrite the expression.

$\frac{x \cdot - 1}{4} - \frac{7}{4} < \frac{3}{4}$

Step 3.5.3

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

$\frac{- 1 \cdot x}{4} - \frac{7}{4} < \frac{3}{4}$

Step 3.5.4

Move the negative in front of the fraction.

$- \frac{x}{4} - \frac{7}{4} < \frac{3}{4}$

Step 4

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

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Step 4.1

Add $\frac{7}{4}$ to both sides of the inequality.

$- \frac{x}{4} < \frac{3}{4} + \frac{7}{4}$

Step 4.2

Combine the numerators over the common denominator.

$- \frac{x}{4} < \frac{3 + 7}{4}$

Step 4.3

Add $3$ and $7$ .

$- \frac{x}{4} < \frac{10}{4}$

Step 4.4

Cancel the common factor of $10$ and $4$ .

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Step 4.4.1

Factor $2$ out of $10$ .

$- \frac{x}{4} < \frac{2 (5)}{4}$

Step 4.4.2

Cancel the common factors.

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Step 4.4.2.1

Factor $2$ out of $4$ .

$- \frac{x}{4} < \frac{2 \cdot 5}{2 \cdot 2}$

Step 4.4.2.2

Cancel the common factor.

$- \frac{x}{4} < \frac{2 \cdot 5}{2 \cdot 2}$

Step 4.4.2.3

Rewrite the expression.

$- \frac{x}{4} < \frac{5}{2}$

Step 5

Divide each term in $- \frac{x}{4} < \frac{5}{2}$ by $- 1$ and simplify.

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Step 5.1

Divide each term in $- \frac{x}{4} < \frac{5}{2}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{x}{4}}{- 1} > \frac{\frac{5}{2}}{- 1}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{x}{4}}{1} > \frac{\frac{5}{2}}{- 1}$

Step 5.2.2

Divide $\frac{x}{4}$ by $1$ .

$\frac{x}{4} > \frac{\frac{5}{2}}{- 1}$

Step 5.3

Simplify the right side.

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Step 5.3.1

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

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

Step 5.3.2

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

$\frac{x}{4} > - \frac{5}{2}$

Step 6

Multiply both sides by $4$ .

$\frac{x}{4} \cdot 4 > - \frac{5}{2} \cdot 4$

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 $4$ .

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Step 7.1.1.1

Cancel the common factor.

$\frac{x}{4} \cdot 4 > - \frac{5}{2} \cdot 4$

Step 7.1.1.2

Rewrite the expression.

$x > - \frac{5}{2} \cdot 4$

Step 7.2

Simplify the right side.

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Step 7.2.1

Simplify $- \frac{5}{2} \cdot 4$ .

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Step 7.2.1.1

Cancel the common factor of $2$ .

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Step 7.2.1.1.1

Move the leading negative in $- \frac{5}{2}$ into the numerator.

$x > \frac{- 5}{2} \cdot 4$

Step 7.2.1.1.2

Factor $2$ out of $4$ .

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

Step 7.2.1.1.3

Cancel the common factor.

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

Step 7.2.1.1.4

Rewrite the expression.

$x > - 5 \cdot 2$

Step 7.2.1.2

Multiply $- 5$ by $2$ .

$x > - 10$

Step 8

Convert the inequality to interval notation.

$(- 10, \infty)$

Step 9