Solve for x 1/5+1/3x>1/2x-1/4

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

Step 1

Combine $\frac{1}{3}$ and $x$ .

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

Step 2

Combine $\frac{1}{2}$ and $x$ .

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

Step 3

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

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

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

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

$\frac{1}{5} + \frac{x}{3} \cdot \frac{2}{2} - \frac{x}{2} > - \frac{1}{4}$

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

$\frac{1}{5} + \frac{x}{3} \cdot \frac{2}{2} - \frac{x}{2} \cdot \frac{3}{3} > - \frac{1}{4}$

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

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

$\frac{1}{5} + \frac{x \cdot 2}{3 \cdot 2} - \frac{x}{2} \cdot \frac{3}{3} > - \frac{1}{4}$

Multiply $3$ by $2$ .

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

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

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

Multiply $2$ by $3$ .

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

Combine the numerators over the common denominator.

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

Simplify each term.

Simplify the numerator.

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

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

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

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

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

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

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

Multiply $- 1$ by $3$ .

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

Subtract $3$ from $2$ .

$\frac{1}{5} + \frac{x \cdot - 1}{6} > - \frac{1}{4}$

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

$\frac{1}{5} + \frac{- 1 \cdot x}{6} > - \frac{1}{4}$

Move the negative in front of the fraction.

$\frac{1}{5} - \frac{x}{6} > - \frac{1}{4}$

Step 4

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

Subtract $\frac{1}{5}$ from both sides of the inequality.

$- \frac{x}{6} > - \frac{1}{4} - \frac{1}{5}$

To write $- \frac{1}{4}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$- \frac{x}{6} > - \frac{1}{4} \cdot \frac{5}{5} - \frac{1}{5}$

To write $- \frac{1}{5}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$- \frac{x}{6} > - \frac{1}{4} \cdot \frac{5}{5} - \frac{1}{5} \cdot \frac{4}{4}$

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

Multiply $\frac{1}{4}$ by $\frac{5}{5}$ .

$- \frac{x}{6} > - \frac{5}{4 \cdot 5} - \frac{1}{5} \cdot \frac{4}{4}$

Multiply $4$ by $5$ .

$- \frac{x}{6} > - \frac{5}{20} - \frac{1}{5} \cdot \frac{4}{4}$

Multiply $\frac{1}{5}$ by $\frac{4}{4}$ .

$- \frac{x}{6} > - \frac{5}{20} - \frac{4}{5 \cdot 4}$

Multiply $5$ by $4$ .

$- \frac{x}{6} > - \frac{5}{20} - \frac{4}{20}$

Combine the numerators over the common denominator.

$- \frac{x}{6} > \frac{- 5 - 4}{20}$

Subtract $4$ from $- 5$ .

$- \frac{x}{6} > \frac{- 9}{20}$

Move the negative in front of the fraction.

$- \frac{x}{6} > - \frac{9}{20}$

Step 5

Divide each term in $- \frac{x}{6} > - \frac{9}{20}$ by $- 1$ and simplify.

Divide each term in $- \frac{x}{6} > - \frac{9}{20}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{x}{6}}{- 1} < \frac{- \frac{9}{20}}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{\frac{x}{6}}{1} < \frac{- \frac{9}{20}}{- 1}$

Divide $\frac{x}{6}$ by $1$ .

$\frac{x}{6} < \frac{- \frac{9}{20}}{- 1}$

Simplify the right side.

Dividing two negative values results in a positive value.

$\frac{x}{6} < \frac{\frac{9}{20}}{1}$

Divide $\frac{9}{20}$ by $1$ .

$\frac{x}{6} < \frac{9}{20}$

Step 6

Multiply both sides by $6$ .

$\frac{x}{6} \cdot 6 < \frac{9}{20} \cdot 6$

Step 7

Simplify.

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{x}{6} \cdot 6 < \frac{9}{20} \cdot 6$

Rewrite the expression.

$x < \frac{9}{20} \cdot 6$

Simplify the right side.

Simplify $\frac{9}{20} \cdot 6$ .

Cancel the common factor of $2$ .

Factor $2$ out of $20$ .

$x < \frac{9}{2 (10)} \cdot 6$

Factor $2$ out of $6$ .

$x < \frac{9}{2 \cdot 10} \cdot (2 \cdot 3)$

Cancel the common factor.

$x < \frac{9}{2 \cdot 10} \cdot (2 \cdot 3)$

Rewrite the expression.

$x < \frac{9}{10} \cdot 3$

Combine $\frac{9}{10}$ and $3$ .

$x < \frac{9 \cdot 3}{10}$

Multiply $9$ by $3$ .

$x < \frac{27}{10}$

Step 8

The result can be shown in multiple forms.

Inequality Form:

$x < \frac{27}{10}$

Interval Notation:

$(- \infty, \frac{27}{10})$

Solve for x 1/5+1/3x>1/2x-1/4

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