Precalculus Examples

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

Step 1

Write $| \frac{1}{4} - \frac{2}{3} x | + \frac{1}{2} > \frac{6}{5}$ as a piecewise.

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Step 1.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$\frac{1}{4} - \frac{2}{3} x \geq 0$

Step 1.2

Solve the inequality.

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

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

$\frac{1}{4} - \frac{x \cdot 2}{3} \geq 0$

Step 1.2.1.2

Move $2$ to the left of $x$ .

$\frac{1}{4} - \frac{2 x}{3} \geq 0$

Step 1.2.2

Subtract $\frac{1}{4}$ from both sides of the inequality.

$- \frac{2 x}{3} \geq - \frac{1}{4}$

Step 1.2.3

Divide each term in $- \frac{2 x}{3} \geq - \frac{1}{4}$ by $- 1$ and simplify.

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Step 1.2.3.1

Divide each term in $- \frac{2 x}{3} \geq - \frac{1}{4}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{2 x}{3}}{- 1} \leq \frac{- \frac{1}{4}}{- 1}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{2 x}{3}}{1} \leq \frac{- \frac{1}{4}}{- 1}$

Step 1.2.3.2.2

Divide $\frac{2 x}{3}$ by $1$ .

$\frac{2 x}{3} \leq \frac{- \frac{1}{4}}{- 1}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Dividing two negative values results in a positive value.

$\frac{2 x}{3} \leq \frac{\frac{1}{4}}{1}$

Step 1.2.3.3.2

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

$\frac{2 x}{3} \leq \frac{1}{4}$

Step 1.2.4

Multiply both sides by $3$ .

$\frac{2 x}{3} \cdot 3 \leq \frac{1}{4} \cdot 3$

Step 1.2.5

Simplify.

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Step 1.2.5.1

Simplify the left side.

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Step 1.2.5.1.1

Cancel the common factor of $3$ .

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Step 1.2.5.1.1.1

Cancel the common factor.

$\frac{2 x}{3} \cdot 3 \leq \frac{1}{4} \cdot 3$

Step 1.2.5.1.1.2

Rewrite the expression.

$2 x \leq \frac{1}{4} \cdot 3$

Step 1.2.5.2

Simplify the right side.

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Step 1.2.5.2.1

Combine $\frac{1}{4}$ and $3$ .

$2 x \leq \frac{3}{4}$

Step 1.2.6

Divide each term in $2 x \leq \frac{3}{4}$ by $2$ and simplify.

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Step 1.2.6.1

Divide each term in $2 x \leq \frac{3}{4}$ by $2$ .

$\frac{2 x}{2} \leq \frac{\frac{3}{4}}{2}$

Step 1.2.6.2

Simplify the left side.

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Step 1.2.6.2.1

Cancel the common factor of $2$ .

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Step 1.2.6.2.1.1

Cancel the common factor.

$\frac{2 x}{2} \leq \frac{\frac{3}{4}}{2}$

Step 1.2.6.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{\frac{3}{4}}{2}$

Step 1.2.6.3

Simplify the right side.

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Step 1.2.6.3.1

Multiply the numerator by the reciprocal of the denominator.

$x \leq \frac{3}{4} \cdot \frac{1}{2}$

Step 1.2.6.3.2

Multiply $\frac{3}{4} \cdot \frac{1}{2}$ .

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Step 1.2.6.3.2.1

Multiply $\frac{3}{4}$ by $\frac{1}{2}$ .

$x \leq \frac{3}{4 \cdot 2}$

Step 1.2.6.3.2.2

Multiply $4$ by $2$ .

$x \leq \frac{3}{8}$

Step 1.3

In the piece where $\frac{1}{4} - \frac{2}{3} x$ is non-negative, remove the absolute value.

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

Step 1.4

To find the interval for the second piece, find where the inside of the absolute value is negative.

$\frac{1}{4} - \frac{2}{3} x < 0$

Step 1.5

Solve the inequality.

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Step 1.5.1

Simplify each term.

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Step 1.5.1.1

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

$\frac{1}{4} - \frac{x \cdot 2}{3} < 0$

Step 1.5.1.2

Move $2$ to the left of $x$ .

$\frac{1}{4} - \frac{2 x}{3} < 0$

Step 1.5.2

Subtract $\frac{1}{4}$ from both sides of the inequality.

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

Step 1.5.3

Divide each term in $- \frac{2 x}{3} < - \frac{1}{4}$ by $- 1$ and simplify.

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Step 1.5.3.1

Divide each term in $- \frac{2 x}{3} < - \frac{1}{4}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

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

Step 1.5.3.2

Simplify the left side.

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Step 1.5.3.2.1

Dividing two negative values results in a positive value.

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

Step 1.5.3.2.2

Divide $\frac{2 x}{3}$ by $1$ .

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

Step 1.5.3.3

Simplify the right side.

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Step 1.5.3.3.1

Dividing two negative values results in a positive value.

$\frac{2 x}{3} > \frac{\frac{1}{4}}{1}$

Step 1.5.3.3.2

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

$\frac{2 x}{3} > \frac{1}{4}$

Step 1.5.4

Multiply both sides by $3$ .

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

Step 1.5.5

Simplify.

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Step 1.5.5.1

Simplify the left side.

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Step 1.5.5.1.1

Cancel the common factor of $3$ .

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Step 1.5.5.1.1.1

Cancel the common factor.

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

Step 1.5.5.1.1.2

Rewrite the expression.

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

Step 1.5.5.2

Simplify the right side.

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Step 1.5.5.2.1

Combine $\frac{1}{4}$ and $3$ .

$2 x > \frac{3}{4}$

Step 1.5.6

Divide each term in $2 x > \frac{3}{4}$ by $2$ and simplify.

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Step 1.5.6.1

Divide each term in $2 x > \frac{3}{4}$ by $2$ .

$\frac{2 x}{2} > \frac{\frac{3}{4}}{2}$

Step 1.5.6.2

Simplify the left side.

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Step 1.5.6.2.1

Cancel the common factor of $2$ .

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Step 1.5.6.2.1.1

Cancel the common factor.

$\frac{2 x}{2} > \frac{\frac{3}{4}}{2}$

Step 1.5.6.2.1.2

Divide $x$ by $1$ .

$x > \frac{\frac{3}{4}}{2}$

Step 1.5.6.3

Simplify the right side.

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Step 1.5.6.3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.5.6.3.2

Multiply $\frac{3}{4} \cdot \frac{1}{2}$ .

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Step 1.5.6.3.2.1

Multiply $\frac{3}{4}$ by $\frac{1}{2}$ .

$x > \frac{3}{4 \cdot 2}$

Step 1.5.6.3.2.2

Multiply $4$ by $2$ .

$x > \frac{3}{8}$

Step 1.6

In the piece where $\frac{1}{4} - \frac{2}{3} x$ is negative, remove the absolute value and multiply by $- 1$ .

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

Step 1.7

Write as a piecewise.

${\begin{cases} \frac{1}{4} - \frac{2}{3} x + \frac{1}{2} > \frac{6}{5} & x \leq \frac{3}{8} \\ - (\frac{1}{4} - \frac{2}{3} x) + \frac{1}{2} > \frac{6}{5} & x > \frac{3}{8} \end{cases}$

Step 1.8

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

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Step 1.8.1

Simplify each term.

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Step 1.8.1.1

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

${\begin{cases} \frac{1}{4} - \frac{x \cdot 2}{3} + \frac{1}{2} > \frac{6}{5} & x \leq \frac{3}{8} \\ - (\frac{1}{4} - \frac{2}{3} x) + \frac{1}{2} > \frac{6}{5} & x > \frac{3}{8} \end{cases}$

Step 1.8.1.2

Move $2$ to the left of $x$ .

${\begin{cases} \frac{1}{4} - \frac{2 x}{3} + \frac{1}{2} > \frac{6}{5} & x \leq \frac{3}{8} \\ - (\frac{1}{4} - \frac{2}{3} x) + \frac{1}{2} > \frac{6}{5} & x > \frac{3}{8} \end{cases}$

Step 1.8.2

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

${\begin{cases} - \frac{2 x}{3} + \frac{1}{4} + \frac{1}{2} \cdot \frac{2}{2} > \frac{6}{5} & x \leq \frac{3}{8} \\ - (\frac{1}{4} - \frac{2}{3} x) + \frac{1}{2} > \frac{6}{5} & x > \frac{3}{8} \end{cases}$

Step 1.8.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.8.3.1

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

${\begin{cases} - \frac{2 x}{3} + \frac{1}{4} + \frac{2}{2 \cdot 2} > \frac{6}{5} & x \leq \frac{3}{8} \\ - (\frac{1}{4} - \frac{2}{3} x) + \frac{1}{2} > \frac{6}{5} & x > \frac{3}{8} \end{cases}$

Step 1.8.3.2

Multiply $2$ by $2$ .

${\begin{cases} - \frac{2 x}{3} + \frac{1}{4} + \frac{2}{4} > \frac{6}{5} & x \leq \frac{3}{8} \\ - (\frac{1}{4} - \frac{2}{3} x) + \frac{1}{2} > \frac{6}{5} & x > \frac{3}{8} \end{cases}$

Step 1.8.4

Combine the numerators over the common denominator.

${\begin{cases} - \frac{2 x}{3} + \frac{1 + 2}{4} > \frac{6}{5} & x \leq \frac{3}{8} \\ - (\frac{1}{4} - \frac{2}{3} x) + \frac{1}{2} > \frac{6}{5} & x > \frac{3}{8} \end{cases}$

Step 1.8.5

Add $1$ and $2$ .

${\begin{cases} - \frac{2 x}{3} + \frac{3}{4} > \frac{6}{5} & x \leq \frac{3}{8} \\ - (\frac{1}{4} - \frac{2}{3} x) + \frac{1}{2} > \frac{6}{5} & x > \frac{3}{8} \end{cases}$

Step 1.9

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

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Step 1.9.1

Simplify each term.

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Step 1.9.1.1

Simplify each term.

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Step 1.9.1.1.1

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

${\begin{cases} - \frac{2 x}{3} + \frac{3}{4} > \frac{6}{5} & x \leq \frac{3}{8} \\ - (\frac{1}{4} - \frac{x \cdot 2}{3}) + \frac{1}{2} > \frac{6}{5} & x > \frac{3}{8} \end{cases}$

Step 1.9.1.1.2

Move $2$ to the left of $x$ .

${\begin{cases} - \frac{2 x}{3} + \frac{3}{4} > \frac{6}{5} & x \leq \frac{3}{8} \\ - (\frac{1}{4} - \frac{2 x}{3}) + \frac{1}{2} > \frac{6}{5} & x > \frac{3}{8} \end{cases}$

Step 1.9.1.2

Apply the distributive property.

${\begin{cases} - \frac{2 x}{3} + \frac{3}{4} > \frac{6}{5} & x \leq \frac{3}{8} \\ - \frac{1}{4} - - \frac{2 x}{3} + \frac{1}{2} > \frac{6}{5} & x > \frac{3}{8} \end{cases}$

Step 1.9.1.3

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

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Step 1.9.1.3.1

Multiply $- 1$ by $- 1$ .

${\begin{cases} - \frac{2 x}{3} + \frac{3}{4} > \frac{6}{5} & x \leq \frac{3}{8} \\ - \frac{1}{4} + 1 \frac{2 x}{3} + \frac{1}{2} > \frac{6}{5} & x > \frac{3}{8} \end{cases}$

Step 1.9.1.3.2

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

${\begin{cases} - \frac{2 x}{3} + \frac{3}{4} > \frac{6}{5} & x \leq \frac{3}{8} \\ - \frac{1}{4} + \frac{2 x}{3} + \frac{1}{2} > \frac{6}{5} & x > \frac{3}{8} \end{cases}$

Step 1.9.2

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

${\begin{cases} - \frac{2 x}{3} + \frac{3}{4} > \frac{6}{5} & x \leq \frac{3}{8} \\ \frac{2 x}{3} - \frac{1}{4} + \frac{1}{2} \cdot \frac{2}{2} > \frac{6}{5} & x > \frac{3}{8} \end{cases}$

Step 1.9.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.9.3.1

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

${\begin{cases} - \frac{2 x}{3} + \frac{3}{4} > \frac{6}{5} & x \leq \frac{3}{8} \\ \frac{2 x}{3} - \frac{1}{4} + \frac{2}{2 \cdot 2} > \frac{6}{5} & x > \frac{3}{8} \end{cases}$

Step 1.9.3.2

Multiply $2$ by $2$ .

${\begin{cases} - \frac{2 x}{3} + \frac{3}{4} > \frac{6}{5} & x \leq \frac{3}{8} \\ \frac{2 x}{3} - \frac{1}{4} + \frac{2}{4} > \frac{6}{5} & x > \frac{3}{8} \end{cases}$

Step 1.9.4

Combine the numerators over the common denominator.

${\begin{cases} - \frac{2 x}{3} + \frac{3}{4} > \frac{6}{5} & x \leq \frac{3}{8} \\ \frac{2 x}{3} + \frac{- 1 + 2}{4} > \frac{6}{5} & x > \frac{3}{8} \end{cases}$

Step 1.9.5

Add $- 1$ and $2$ .

${\begin{cases} - \frac{2 x}{3} + \frac{3}{4} > \frac{6}{5} & x \leq \frac{3}{8} \\ \frac{2 x}{3} + \frac{1}{4} > \frac{6}{5} & x > \frac{3}{8} \end{cases}$

Step 2

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

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Step 2.1

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

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Step 2.1.1

Subtract $\frac{3}{4}$ from both sides of the inequality.

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

Step 2.1.2

To write $\frac{6}{5}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Step 2.1.3

To write $- \frac{3}{4}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

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

Step 2.1.4

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

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Step 2.1.4.1

Multiply $\frac{6}{5}$ by $\frac{4}{4}$ .

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

Step 2.1.4.2

Multiply $5$ by $4$ .

$- \frac{2 x}{3} > \frac{6 \cdot 4}{20} - \frac{3}{4} \cdot \frac{5}{5}$

Step 2.1.4.3

Multiply $\frac{3}{4}$ by $\frac{5}{5}$ .

$- \frac{2 x}{3} > \frac{6 \cdot 4}{20} - \frac{3 \cdot 5}{4 \cdot 5}$

Step 2.1.4.4

Multiply $4$ by $5$ .

$- \frac{2 x}{3} > \frac{6 \cdot 4}{20} - \frac{3 \cdot 5}{20}$

Step 2.1.5

Combine the numerators over the common denominator.

$- \frac{2 x}{3} > \frac{6 \cdot 4 - 3 \cdot 5}{20}$

Step 2.1.6

Simplify the numerator.

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Step 2.1.6.1

Multiply $6$ by $4$ .

$- \frac{2 x}{3} > \frac{24 - 3 \cdot 5}{20}$

Step 2.1.6.2

Multiply $- 3$ by $5$ .

$- \frac{2 x}{3} > \frac{24 - 15}{20}$

Step 2.1.6.3

Subtract $15$ from $24$ .

$- \frac{2 x}{3} > \frac{9}{20}$

Step 2.2

Divide each term in $- \frac{2 x}{3} > \frac{9}{20}$ by $- 1$ and simplify.

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Step 2.2.1

Divide each term in $- \frac{2 x}{3} > \frac{9}{20}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{2 x}{3}}{- 1} < \frac{\frac{9}{20}}{- 1}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{2 x}{3}}{1} < \frac{\frac{9}{20}}{- 1}$

Step 2.2.2.2

Divide $\frac{2 x}{3}$ by $1$ .

$\frac{2 x}{3} < \frac{\frac{9}{20}}{- 1}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Move the negative one from the denominator of $\frac{\frac{9}{20}}{- 1}$ .

$\frac{2 x}{3} < - 1 \cdot \frac{9}{20}$

Step 2.2.3.2

Rewrite $- 1 \cdot \frac{9}{20}$ as $- \frac{9}{20}$ .

$\frac{2 x}{3} < - \frac{9}{20}$

Step 2.3

Multiply both sides by $3$ .

$\frac{2 x}{3} \cdot 3 < - \frac{9}{20} \cdot 3$

Step 2.4

Simplify.

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Step 2.4.1

Simplify the left side.

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Step 2.4.1.1

Cancel the common factor of $3$ .

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Step 2.4.1.1.1

Cancel the common factor.

$\frac{2 x}{3} \cdot 3 < - \frac{9}{20} \cdot 3$

Step 2.4.1.1.2

Rewrite the expression.

$2 x < - \frac{9}{20} \cdot 3$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Simplify $- \frac{9}{20} \cdot 3$ .

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Step 2.4.2.1.1

Multiply $- \frac{9}{20} \cdot 3$ .

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Step 2.4.2.1.1.1

Multiply $3$ by $- 1$ .

$2 x < - 3 (\frac{9}{20})$

Step 2.4.2.1.1.2

Combine $- 3$ and $\frac{9}{20}$ .

$2 x < \frac{- 3 \cdot 9}{20}$

Step 2.4.2.1.1.3

Multiply $- 3$ by $9$ .

$2 x < \frac{- 27}{20}$

Step 2.4.2.1.2

Move the negative in front of the fraction.

$2 x < - \frac{27}{20}$

Step 2.5

Divide each term in $2 x < - \frac{27}{20}$ by $2$ and simplify.

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Step 2.5.1

Divide each term in $2 x < - \frac{27}{20}$ by $2$ .

$\frac{2 x}{2} < \frac{- \frac{27}{20}}{2}$

Step 2.5.2

Simplify the left side.

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Step 2.5.2.1

Cancel the common factor of $2$ .

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Step 2.5.2.1.1

Cancel the common factor.

$\frac{2 x}{2} < \frac{- \frac{27}{20}}{2}$

Step 2.5.2.1.2

Divide $x$ by $1$ .

$x < \frac{- \frac{27}{20}}{2}$

Step 2.5.3

Simplify the right side.

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Step 2.5.3.1

Multiply the numerator by the reciprocal of the denominator.

$x < - \frac{27}{20} \cdot \frac{1}{2}$

Step 2.5.3.2

Multiply $- \frac{27}{20} \cdot \frac{1}{2}$ .

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Step 2.5.3.2.1

Multiply $\frac{1}{2}$ by $\frac{27}{20}$ .

$x < - \frac{27}{2 \cdot 20}$

Step 2.5.3.2.2

Multiply $2$ by $20$ .

$x < - \frac{27}{40}$

Step 3

Solve $\frac{2 x}{3} + \frac{1}{4} > \frac{6}{5}$ for $x$ .

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Step 3.1

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

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Step 3.1.1

Subtract $\frac{1}{4}$ from both sides of the inequality.

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

Step 3.1.2

To write $\frac{6}{5}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Step 3.1.3

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

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

Step 3.1.4

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

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Step 3.1.4.1

Multiply $\frac{6}{5}$ by $\frac{4}{4}$ .

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

Step 3.1.4.2

Multiply $5$ by $4$ .

$\frac{2 x}{3} > \frac{6 \cdot 4}{20} - \frac{1}{4} \cdot \frac{5}{5}$

Step 3.1.4.3

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

$\frac{2 x}{3} > \frac{6 \cdot 4}{20} - \frac{5}{4 \cdot 5}$

Step 3.1.4.4

Multiply $4$ by $5$ .

$\frac{2 x}{3} > \frac{6 \cdot 4}{20} - \frac{5}{20}$

Step 3.1.5

Combine the numerators over the common denominator.

$\frac{2 x}{3} > \frac{6 \cdot 4 - 5}{20}$

Step 3.1.6

Simplify the numerator.

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Step 3.1.6.1

Multiply $6$ by $4$ .

$\frac{2 x}{3} > \frac{24 - 5}{20}$

Step 3.1.6.2

Subtract $5$ from $24$ .

$\frac{2 x}{3} > \frac{19}{20}$

Step 3.2

Multiply both sides by $3$ .

$\frac{2 x}{3} \cdot 3 > \frac{19}{20} \cdot 3$

Step 3.3

Simplify.

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Step 3.3.1

Simplify the left side.

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Step 3.3.1.1

Cancel the common factor of $3$ .

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Step 3.3.1.1.1

Cancel the common factor.

$\frac{2 x}{3} \cdot 3 > \frac{19}{20} \cdot 3$

Step 3.3.1.1.2

Rewrite the expression.

$2 x > \frac{19}{20} \cdot 3$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Multiply $\frac{19}{20} \cdot 3$ .

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Step 3.3.2.1.1

Combine $\frac{19}{20}$ and $3$ .

$2 x > \frac{19 \cdot 3}{20}$

Step 3.3.2.1.2

Multiply $19$ by $3$ .

$2 x > \frac{57}{20}$

Step 3.4

Divide each term in $2 x > \frac{57}{20}$ by $2$ and simplify.

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Step 3.4.1

Divide each term in $2 x > \frac{57}{20}$ by $2$ .

$\frac{2 x}{2} > \frac{\frac{57}{20}}{2}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $2$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{2 x}{2} > \frac{\frac{57}{20}}{2}$

Step 3.4.2.1.2

Divide $x$ by $1$ .

$x > \frac{\frac{57}{20}}{2}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Multiply the numerator by the reciprocal of the denominator.

$x > \frac{57}{20} \cdot \frac{1}{2}$

Step 3.4.3.2

Multiply $\frac{57}{20} \cdot \frac{1}{2}$ .

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Step 3.4.3.2.1

Multiply $\frac{57}{20}$ by $\frac{1}{2}$ .

$x > \frac{57}{20 \cdot 2}$

Step 3.4.3.2.2

Multiply $20$ by $2$ .

$x > \frac{57}{40}$

Step 4

Find the union of the solutions.

$x < - \frac{27}{40}$ or $x > \frac{57}{40}$

Step 5

Convert the inequality to interval notation.

$(- \infty, - \frac{27}{40}) \cup (\frac{57}{40}, \infty)$

Step 6