Algebra Examples

$\frac{1}{3} < - x - \frac{5}{6} < \frac{2}{3} x + 4$

Step 1

Solve $\frac{1}{3} < - x - \frac{5}{6}$ for $x$ .

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Step 1.1

Rewrite so $x$ is on the left side of the inequality.

$- x - \frac{5}{6} > \frac{1}{3}$

Step 1.2

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

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Step 1.2.1

Add $\frac{5}{6}$ to both sides of the inequality.

$- x > \frac{1}{3} + \frac{5}{6}$

Step 1.2.2

To write $\frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- x > \frac{1}{3} \cdot \frac{2}{2} + \frac{5}{6}$

Step 1.2.3

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

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Step 1.2.3.1

Multiply $\frac{1}{3}$ by $\frac{2}{2}$ .

$- x > \frac{2}{3 \cdot 2} + \frac{5}{6}$

Step 1.2.3.2

Multiply $3$ by $2$ .

$- x > \frac{2}{6} + \frac{5}{6}$

Step 1.2.4

Combine the numerators over the common denominator.

$- x > \frac{2 + 5}{6}$

Step 1.2.5

Add $2$ and $5$ .

$- x > \frac{7}{6}$

Step 1.3

Divide each term in $- x > \frac{7}{6}$ by $- 1$ and simplify.

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Step 1.3.1

Divide each term in $- x > \frac{7}{6}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} < \frac{\frac{7}{6}}{- 1}$

Step 1.3.2

Simplify the left side.

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Step 1.3.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} < \frac{\frac{7}{6}}{- 1}$

Step 1.3.2.2

Divide $x$ by $1$ .

$x < \frac{\frac{7}{6}}{- 1}$

Step 1.3.3

Simplify the right side.

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Step 1.3.3.1

Move the negative one from the denominator of $\frac{\frac{7}{6}}{- 1}$ .

$x < - 1 \cdot \frac{7}{6}$

Step 1.3.3.2

Rewrite $- 1 \cdot \frac{7}{6}$ as $- \frac{7}{6}$ .

$x < - \frac{7}{6}$

Step 2

Solve $- x - \frac{5}{6} < \frac{2}{3} x + 4$ for $x$ .

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Step 2.1

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

$- x - \frac{5}{6} < \frac{2 x}{3} + 4$

Step 2.2

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

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Step 2.2.1

Subtract $\frac{2 x}{3}$ from both sides of the inequality.

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

Combine the numerators over the common denominator.

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

Step 2.2.5

Simplify each term.

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Step 2.2.5.1

Simplify the numerator.

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Step 2.2.5.1.1

Factor $x$ out of $- x \cdot 3 - 2 x$ .

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Step 2.2.5.1.1.1

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

$\frac{x (- 1 \cdot 3) - 2 x}{3} - \frac{5}{6} < 4$

Step 2.2.5.1.1.2

Factor $x$ out of $- 2 x$ .

$\frac{x (- 1 \cdot 3) + x \cdot - 2}{3} - \frac{5}{6} < 4$

Step 2.2.5.1.1.3

Factor $x$ out of $x (- 1 \cdot 3) + x \cdot - 2$ .

$\frac{x (- 1 \cdot 3 - 2)}{3} - \frac{5}{6} < 4$

Step 2.2.5.1.2

Multiply $- 1$ by $3$ .

$\frac{x (- 3 - 2)}{3} - \frac{5}{6} < 4$

Step 2.2.5.1.3

Subtract $2$ from $- 3$ .

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

Step 2.2.5.2

Move $- 5$ to the left of $x$ .

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

Step 2.2.5.3

Move the negative in front of the fraction.

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

Step 2.3

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

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Step 2.3.1

Add $\frac{5}{6}$ to both sides of the inequality.

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

Step 2.3.2

To write $4$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

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

Step 2.3.3

Combine $4$ and $\frac{6}{6}$ .

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

Step 2.3.4

Combine the numerators over the common denominator.

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

Step 2.3.5

Simplify the numerator.

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Step 2.3.5.1

Multiply $4$ by $6$ .

$- \frac{5 x}{3} < \frac{24 + 5}{6}$

Step 2.3.5.2

Add $24$ and $5$ .

$- \frac{5 x}{3} < \frac{29}{6}$

Step 2.4

Divide each term in $- \frac{5 x}{3} < \frac{29}{6}$ by $- 1$ and simplify.

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Step 2.4.1

Divide each term in $- \frac{5 x}{3} < \frac{29}{6}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{5 x}{3}}{- 1} > \frac{\frac{29}{6}}{- 1}$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{5 x}{3}}{1} > \frac{\frac{29}{6}}{- 1}$

Step 2.4.2.2

Divide $\frac{5 x}{3}$ by $1$ .

$\frac{5 x}{3} > \frac{\frac{29}{6}}{- 1}$

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Move the negative one from the denominator of $\frac{\frac{29}{6}}{- 1}$ .

$\frac{5 x}{3} > - 1 \cdot \frac{29}{6}$

Step 2.4.3.2

Rewrite $- 1 \cdot \frac{29}{6}$ as $- \frac{29}{6}$ .

$\frac{5 x}{3} > - \frac{29}{6}$

Step 2.5

Multiply both sides by $3$ .

$\frac{5 x}{3} \cdot 3 > - \frac{29}{6} \cdot 3$

Step 2.6

Simplify.

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Step 2.6.1

Simplify the left side.

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Step 2.6.1.1

Cancel the common factor of $3$ .

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Step 2.6.1.1.1

Cancel the common factor.

$\frac{5 x}{3} \cdot 3 > - \frac{29}{6} \cdot 3$

Step 2.6.1.1.2

Rewrite the expression.

$5 x > - \frac{29}{6} \cdot 3$

Step 2.6.2

Simplify the right side.

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Step 2.6.2.1

Simplify $- \frac{29}{6} \cdot 3$ .

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Step 2.6.2.1.1

Cancel the common factor of $3$ .

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Step 2.6.2.1.1.1

Move the leading negative in $- \frac{29}{6}$ into the numerator.

$5 x > \frac{- 29}{6} \cdot 3$

Step 2.6.2.1.1.2

Factor $3$ out of $6$ .

$5 x > \frac{- 29}{3 (2)} \cdot 3$

Step 2.6.2.1.1.3

Cancel the common factor.

$5 x > \frac{- 29}{3 \cdot 2} \cdot 3$

Step 2.6.2.1.1.4

Rewrite the expression.

$5 x > \frac{- 29}{2}$

Step 2.6.2.1.2

Move the negative in front of the fraction.

$5 x > - \frac{29}{2}$

Step 2.7

Divide each term in $5 x > - \frac{29}{2}$ by $5$ and simplify.

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Step 2.7.1

Divide each term in $5 x > - \frac{29}{2}$ by $5$ .

$\frac{5 x}{5} > \frac{- \frac{29}{2}}{5}$

Step 2.7.2

Simplify the left side.

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Step 2.7.2.1

Cancel the common factor of $5$ .

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Step 2.7.2.1.1

Cancel the common factor.

$\frac{5 x}{5} > \frac{- \frac{29}{2}}{5}$

Step 2.7.2.1.2

Divide $x$ by $1$ .

$x > \frac{- \frac{29}{2}}{5}$

Step 2.7.3

Simplify the right side.

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Step 2.7.3.1

Multiply the numerator by the reciprocal of the denominator.

$x > - \frac{29}{2} \cdot \frac{1}{5}$

Step 2.7.3.2

Multiply $- \frac{29}{2} \cdot \frac{1}{5}$ .

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Step 2.7.3.2.1

Multiply $\frac{1}{5}$ by $\frac{29}{2}$ .

$x > - \frac{29}{5 \cdot 2}$

Step 2.7.3.2.2

Multiply $5$ by $2$ .

$x > - \frac{29}{10}$

Step 3

Find the intersection of $x < - \frac{7}{6}$ and $x > - \frac{29}{10}$ .

$- \frac{29}{10} < x < - \frac{7}{6}$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$- \frac{29}{10} < x < - \frac{7}{6}$

Interval Notation:

$(- \frac{29}{10}, - \frac{7}{6})$

Step 5