Solve the Inequality for x 1/3<-x-5/6<2/3x+4

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

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

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

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

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

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

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

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

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

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

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

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

Multiply $3$ by $2$ .

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

Combine the numerators over the common denominator.

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

Add $2$ and $5$ .

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

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

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.

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

Simplify the left side.

Dividing two negative values results in a positive value.

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

Divide $x$ by $1$ .

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

Simplify the right side.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify each term.

Simplify the numerator.

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

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

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

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

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

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

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

Multiply $- 1$ by $3$ .

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

Subtract $2$ from $- 3$ .

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

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

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

Move the negative in front of the fraction.

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

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $4$ by $6$ .

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

Add $24$ and $5$ .

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

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

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.

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

Simplify the left side.

Dividing two negative values results in a positive value.

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

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

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

Simplify the right side.

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

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

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

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

Multiply both sides by $3$ .

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

Simplify.

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify the right side.

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

Cancel the common factor of $3$ .

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

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

Factor $3$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative in front of the fraction.

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

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

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

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

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

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

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

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

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

Multiply $5$ by $2$ .

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

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

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

The result can be shown in multiple forms.

Inequality Form:

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

Interval Notation:

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

Solve the Inequality for x 1/3<-x-5/6<2/3x+4

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