Finite Math Examples

$\frac{7}{3} + \frac{2}{3} x < \frac{49}{18} + \frac{7}{6} x + \frac{3}{6}$

Step 1

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

$\frac{7}{3} + \frac{2 x}{3} < \frac{49}{18} + \frac{7}{6} x + \frac{3}{6}$

Step 2

Simplify $\frac{49}{18} + \frac{7}{6} x + \frac{3}{6}$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Combine $\frac{7}{6}$ and $x$ .

$\frac{7}{3} + \frac{2 x}{3} < \frac{49}{18} + \frac{7 x}{6} + \frac{3}{6}$

Step 2.1.2

Cancel the common factor of $3$ and $6$ .

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Step 2.1.2.1

Factor $3$ out of $3$ .

$\frac{7}{3} + \frac{2 x}{3} < \frac{49}{18} + \frac{7 x}{6} + \frac{3 (1)}{6}$

Step 2.1.2.2

Cancel the common factors.

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Step 2.1.2.2.1

Factor $3$ out of $6$ .

$\frac{7}{3} + \frac{2 x}{3} < \frac{49}{18} + \frac{7 x}{6} + \frac{3 \cdot 1}{3 \cdot 2}$

Step 2.1.2.2.2

Cancel the common factor.

$\frac{7}{3} + \frac{2 x}{3} < \frac{49}{18} + \frac{7 x}{6} + \frac{3 \cdot 1}{3 \cdot 2}$

Step 2.1.2.2.3

Rewrite the expression.

$\frac{7}{3} + \frac{2 x}{3} < \frac{49}{18} + \frac{7 x}{6} + \frac{1}{2}$

Step 2.2

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\frac{7}{3} + \frac{2 x}{3} < \frac{7 x}{6} + \frac{49}{18} + \frac{1}{2} \cdot \frac{9}{9}$

Step 2.3

Write each expression with a common denominator of $18$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

Multiply $\frac{1}{2}$ by $\frac{9}{9}$ .

$\frac{7}{3} + \frac{2 x}{3} < \frac{7 x}{6} + \frac{49}{18} + \frac{9}{2 \cdot 9}$

Step 2.3.2

Multiply $2$ by $9$ .

$\frac{7}{3} + \frac{2 x}{3} < \frac{7 x}{6} + \frac{49}{18} + \frac{9}{18}$

Step 2.4

Simplify the expression.

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Step 2.4.1

Combine the numerators over the common denominator.

$\frac{7}{3} + \frac{2 x}{3} < \frac{7 x}{6} + \frac{49 + 9}{18}$

Step 2.4.2

Add $49$ and $9$ .

$\frac{7}{3} + \frac{2 x}{3} < \frac{7 x}{6} + \frac{58}{18}$

Step 2.5

Cancel the common factor of $58$ and $18$ .

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Step 2.5.1

Factor $2$ out of $58$ .

$\frac{7}{3} + \frac{2 x}{3} < \frac{7 x}{6} + \frac{2 (29)}{18}$

Step 2.5.2

Cancel the common factors.

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Step 2.5.2.1

Factor $2$ out of $18$ .

$\frac{7}{3} + \frac{2 x}{3} < \frac{7 x}{6} + \frac{2 \cdot 29}{2 \cdot 9}$

Step 2.5.2.2

Cancel the common factor.

$\frac{7}{3} + \frac{2 x}{3} < \frac{7 x}{6} + \frac{2 \cdot 29}{2 \cdot 9}$

Step 2.5.2.3

Rewrite the expression.

$\frac{7}{3} + \frac{2 x}{3} < \frac{7 x}{6} + \frac{29}{9}$

Step 3

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

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Step 3.1

Subtract $\frac{7 x}{6}$ from both sides of the inequality.

$\frac{7}{3} + \frac{2 x}{3} - \frac{7 x}{6} < \frac{29}{9}$

Step 3.2

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

$\frac{7}{3} + \frac{2 x}{3} \cdot \frac{2}{2} - \frac{7 x}{6} < \frac{29}{9}$

Step 3.3

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.3.1

Multiply $\frac{2 x}{3}$ by $\frac{2}{2}$ .

$\frac{7}{3} + \frac{2 x \cdot 2}{3 \cdot 2} - \frac{7 x}{6} < \frac{29}{9}$

Step 3.3.2

Multiply $3$ by $2$ .

$\frac{7}{3} + \frac{2 x \cdot 2}{6} - \frac{7 x}{6} < \frac{29}{9}$

Step 3.4

Combine the numerators over the common denominator.

$\frac{7}{3} + \frac{2 x \cdot 2 - 7 x}{6} < \frac{29}{9}$

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 $2 x \cdot 2 - 7 x$ .

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Step 3.5.1.1.1

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

$\frac{7}{3} + \frac{x (2 \cdot 2) - 7 x}{6} < \frac{29}{9}$

Step 3.5.1.1.2

Factor $x$ out of $- 7 x$ .

$\frac{7}{3} + \frac{x (2 \cdot 2) + x \cdot - 7}{6} < \frac{29}{9}$

Step 3.5.1.1.3

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

$\frac{7}{3} + \frac{x (2 \cdot 2 - 7)}{6} < \frac{29}{9}$

Step 3.5.1.2

Multiply $2$ by $2$ .

$\frac{7}{3} + \frac{x (4 - 7)}{6} < \frac{29}{9}$

Step 3.5.1.3

Subtract $7$ from $4$ .

$\frac{7}{3} + \frac{x \cdot - 3}{6} < \frac{29}{9}$

Step 3.5.2

Cancel the common factor of $- 3$ and $6$ .

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Step 3.5.2.1

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

$\frac{7}{3} + \frac{3 (x \cdot - 1)}{6} < \frac{29}{9}$

Step 3.5.2.2

Cancel the common factors.

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Step 3.5.2.2.1

Factor $3$ out of $6$ .

$\frac{7}{3} + \frac{3 (x \cdot - 1)}{3 (2)} < \frac{29}{9}$

Step 3.5.2.2.2

Cancel the common factor.

$\frac{7}{3} + \frac{3 (x \cdot - 1)}{3 \cdot 2} < \frac{29}{9}$

Step 3.5.2.2.3

Rewrite the expression.

$\frac{7}{3} + \frac{x \cdot - 1}{2} < \frac{29}{9}$

Step 3.5.3

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

$\frac{7}{3} + \frac{- 1 \cdot x}{2} < \frac{29}{9}$

Step 3.5.4

Move the negative in front of the fraction.

$\frac{7}{3} - \frac{x}{2} < \frac{29}{9}$

Step 4

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

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Step 4.1

Subtract $\frac{7}{3}$ from both sides of the inequality.

$- \frac{x}{2} < \frac{29}{9} - \frac{7}{3}$

Step 4.2

To write $- \frac{7}{3}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{x}{2} < \frac{29}{9} - \frac{7}{3} \cdot \frac{3}{3}$

Step 4.3

Write each expression with a common denominator of $9$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.3.1

Multiply $\frac{7}{3}$ by $\frac{3}{3}$ .

$- \frac{x}{2} < \frac{29}{9} - \frac{7 \cdot 3}{3 \cdot 3}$

Step 4.3.2

Multiply $3$ by $3$ .

$- \frac{x}{2} < \frac{29}{9} - \frac{7 \cdot 3}{9}$

Step 4.4

Combine the numerators over the common denominator.

$- \frac{x}{2} < \frac{29 - 7 \cdot 3}{9}$

Step 4.5

Simplify the numerator.

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Step 4.5.1

Multiply $- 7$ by $3$ .

$- \frac{x}{2} < \frac{29 - 21}{9}$

Step 4.5.2

Subtract $21$ from $29$ .

$- \frac{x}{2} < \frac{8}{9}$

Step 5

Divide each term in $- \frac{x}{2} < \frac{8}{9}$ by $- 1$ and simplify.

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Step 5.1

Divide each term in $- \frac{x}{2} < \frac{8}{9}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{x}{2}}{- 1} > \frac{\frac{8}{9}}{- 1}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{x}{2}}{1} > \frac{\frac{8}{9}}{- 1}$

Step 5.2.2

Divide $\frac{x}{2}$ by $1$ .

$\frac{x}{2} > \frac{\frac{8}{9}}{- 1}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Move the negative one from the denominator of $\frac{\frac{8}{9}}{- 1}$ .

$\frac{x}{2} > - 1 \cdot \frac{8}{9}$

Step 5.3.2

Rewrite $- 1 \cdot \frac{8}{9}$ as $- \frac{8}{9}$ .

$\frac{x}{2} > - \frac{8}{9}$

Step 6

Multiply both sides by $2$ .

$\frac{x}{2} \cdot 2 > - \frac{8}{9} \cdot 2$

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 $2$ .

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Step 7.1.1.1

Cancel the common factor.

$\frac{x}{2} \cdot 2 > - \frac{8}{9} \cdot 2$

Step 7.1.1.2

Rewrite the expression.

$x > - \frac{8}{9} \cdot 2$

Step 7.2

Simplify the right side.

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Step 7.2.1

Simplify $- \frac{8}{9} \cdot 2$ .

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Step 7.2.1.1

Multiply $- \frac{8}{9} \cdot 2$ .

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Step 7.2.1.1.1

Multiply $2$ by $- 1$ .

$x > - 2 (\frac{8}{9})$

Step 7.2.1.1.2

Combine $- 2$ and $\frac{8}{9}$ .

$x > \frac{- 2 \cdot 8}{9}$

Step 7.2.1.1.3

Multiply $- 2$ by $8$ .

$x > \frac{- 16}{9}$

Step 7.2.1.2

Move the negative in front of the fraction.

$x > - \frac{16}{9}$

Step 8

The result can be shown in multiple forms.

Inequality Form:

$x > - \frac{16}{9}$

Interval Notation:

$(- \frac{16}{9}, \infty)$