Finite Math Examples

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

Step 1

Move all the expressions to the left side of the equation.

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Step 1.1

Subtract $\frac{2 x - 1}{2}$ from both sides of the inequality.

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

Step 1.2

Subtract $\frac{6 x + 1}{3 x - 2}$ from both sides of the inequality.

$x - \frac{1}{2 - 3 x} - \frac{2 x - 1}{2} - \frac{6 x + 1}{3 x - 2} > 0$

Step 2

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

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Step 2.1

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

$\frac{x (2 - 3 x)}{2 - 3 x} - \frac{1}{2 - 3 x} - \frac{2 x - 1}{2} - \frac{6 x + 1}{3 x - 2} > 0$

Step 2.2

Combine the numerators over the common denominator.

$\frac{x (2 - 3 x) - 1}{2 - 3 x} - \frac{2 x - 1}{2} - \frac{6 x + 1}{3 x - 2} > 0$

Step 2.3

Simplify the numerator.

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Step 2.3.1

Apply the distributive property.

$\frac{x \cdot 2 + x (- 3 x) - 1}{2 - 3 x} - \frac{2 x - 1}{2} - \frac{6 x + 1}{3 x - 2} > 0$

Step 2.3.2

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

$\frac{2 \cdot x + x (- 3 x) - 1}{2 - 3 x} - \frac{2 x - 1}{2} - \frac{6 x + 1}{3 x - 2} > 0$

Step 2.3.3

Rewrite using the commutative property of multiplication.

$\frac{2 \cdot x - 3 x \cdot x - 1}{2 - 3 x} - \frac{2 x - 1}{2} - \frac{6 x + 1}{3 x - 2} > 0$

Step 2.3.4

Multiply $x$ by $x$ by adding the exponents.

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Step 2.3.4.1

Move $x$ .

$\frac{2 x - 3 (x \cdot x) - 1}{2 - 3 x} - \frac{2 x - 1}{2} - \frac{6 x + 1}{3 x - 2} > 0$

Step 2.3.4.2

Multiply $x$ by $x$ .

$\frac{2 x - 3 x^{2} - 1}{2 - 3 x} - \frac{2 x - 1}{2} - \frac{6 x + 1}{3 x - 2} > 0$

Step 2.3.5

Reorder terms.

$\frac{- 3 x^{2} + 2 x - 1}{2 - 3 x} - \frac{2 x - 1}{2} - \frac{6 x + 1}{3 x - 2} > 0$

Step 2.4

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

$\frac{- 3 x^{2} + 2 x - 1}{2 - 3 x} \cdot \frac{2}{2} - \frac{2 x - 1}{2} - \frac{6 x + 1}{3 x - 2} > 0$

Step 2.5

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

$\frac{- 3 x^{2} + 2 x - 1}{2 - 3 x} \cdot \frac{2}{2} - \frac{2 x - 1}{2} \cdot \frac{2 - 3 x}{2 - 3 x} - \frac{6 x + 1}{3 x - 2} > 0$

Step 2.6

Write each expression with a common denominator of $(2 - 3 x) \cdot 2$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.6.1

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

$\frac{(- 3 x^{2} + 2 x - 1) \cdot 2}{(2 - 3 x) \cdot 2} - \frac{2 x - 1}{2} \cdot \frac{2 - 3 x}{2 - 3 x} - \frac{6 x + 1}{3 x - 2} > 0$

Step 2.6.2

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

$\frac{(- 3 x^{2} + 2 x - 1) \cdot 2}{(2 - 3 x) \cdot 2} - \frac{(2 x - 1) (2 - 3 x)}{2 (2 - 3 x)} - \frac{6 x + 1}{3 x - 2} > 0$

Step 2.6.3

Reorder the factors of $(2 - 3 x) \cdot 2$ .

$\frac{(- 3 x^{2} + 2 x - 1) \cdot 2}{2 (2 - 3 x)} - \frac{(2 x - 1) (2 - 3 x)}{2 (2 - 3 x)} - \frac{6 x + 1}{3 x - 2} > 0$

Step 2.7

Combine the numerators over the common denominator.

$\frac{(- 3 x^{2} + 2 x - 1) \cdot 2 - (2 x - 1) (2 - 3 x)}{2 (2 - 3 x)} - \frac{6 x + 1}{3 x - 2} > 0$

Step 2.8

Simplify each term.

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Step 2.8.1

Simplify the numerator.

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Step 2.8.1.1

Apply the distributive property.

$\frac{- 3 x^{2} \cdot 2 + 2 x \cdot 2 - 1 \cdot 2 - (2 x - 1) (2 - 3 x)}{2 (2 - 3 x)} - \frac{6 x + 1}{3 x - 2} > 0$

Step 2.8.1.2

Simplify.

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Step 2.8.1.2.1

Multiply $2$ by $- 3$ .

$\frac{- 6 x^{2} + 2 x \cdot 2 - 1 \cdot 2 - (2 x - 1) (2 - 3 x)}{2 (2 - 3 x)} - \frac{6 x + 1}{3 x - 2} > 0$

Step 2.8.1.2.2

Multiply $2$ by $2$ .

$\frac{- 6 x^{2} + 4 x - 1 \cdot 2 - (2 x - 1) (2 - 3 x)}{2 (2 - 3 x)} - \frac{6 x + 1}{3 x - 2} > 0$

Step 2.8.1.2.3

Multiply $- 1$ by $2$ .

$\frac{- 6 x^{2} + 4 x - 2 - (2 x - 1) (2 - 3 x)}{2 (2 - 3 x)} - \frac{6 x + 1}{3 x - 2} > 0$

Step 2.8.1.3

Apply the distributive property.

$\frac{- 6 x^{2} + 4 x - 2 + (- (2 x) - - 1) (2 - 3 x)}{2 (2 - 3 x)} - \frac{6 x + 1}{3 x - 2} > 0$

Step 2.8.1.4

Multiply $2$ by $- 1$ .

$\frac{- 6 x^{2} + 4 x - 2 + (- 2 x - - 1) (2 - 3 x)}{2 (2 - 3 x)} - \frac{6 x + 1}{3 x - 2} > 0$

Step 2.8.1.5

Multiply $- 1$ by $- 1$ .

$\frac{- 6 x^{2} + 4 x - 2 + (- 2 x + 1) (2 - 3 x)}{2 (2 - 3 x)} - \frac{6 x + 1}{3 x - 2} > 0$

Step 2.8.1.6

Expand $(- 2 x + 1) (2 - 3 x)$ using the FOIL Method.

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Step 2.8.1.6.1

Apply the distributive property.

$\frac{- 6 x^{2} + 4 x - 2 - 2 x (2 - 3 x) + 1 (2 - 3 x)}{2 (2 - 3 x)} - \frac{6 x + 1}{3 x - 2} > 0$

Step 2.8.1.6.2

Apply the distributive property.

$\frac{- 6 x^{2} + 4 x - 2 - 2 x \cdot 2 - 2 x (- 3 x) + 1 (2 - 3 x)}{2 (2 - 3 x)} - \frac{6 x + 1}{3 x - 2} > 0$

Step 2.8.1.6.3

Apply the distributive property.

$\frac{- 6 x^{2} + 4 x - 2 - 2 x \cdot 2 - 2 x (- 3 x) + 1 \cdot 2 + 1 (- 3 x)}{2 (2 - 3 x)} - \frac{6 x + 1}{3 x - 2} > 0$

Step 2.8.1.7

Simplify and combine like terms.

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Step 2.8.1.7.1

Simplify each term.

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Step 2.8.1.7.1.1

Multiply $2$ by $- 2$ .

$\frac{- 6 x^{2} + 4 x - 2 - 4 x - 2 x (- 3 x) + 1 \cdot 2 + 1 (- 3 x)}{2 (2 - 3 x)} - \frac{6 x + 1}{3 x - 2} > 0$

Step 2.8.1.7.1.2

Rewrite using the commutative property of multiplication.

$\frac{- 6 x^{2} + 4 x - 2 - 4 x - 2 \cdot - 3 x \cdot x + 1 \cdot 2 + 1 (- 3 x)}{2 (2 - 3 x)} - \frac{6 x + 1}{3 x - 2} > 0$

Step 2.8.1.7.1.3

Multiply $x$ by $x$ by adding the exponents.

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Step 2.8.1.7.1.3.1

Move $x$ .

$\frac{- 6 x^{2} + 4 x - 2 - 4 x - 2 \cdot - 3 (x \cdot x) + 1 \cdot 2 + 1 (- 3 x)}{2 (2 - 3 x)} - \frac{6 x + 1}{3 x - 2} > 0$

Step 2.8.1.7.1.3.2

Multiply $x$ by $x$ .

$\frac{- 6 x^{2} + 4 x - 2 - 4 x - 2 \cdot - 3 x^{2} + 1 \cdot 2 + 1 (- 3 x)}{2 (2 - 3 x)} - \frac{6 x + 1}{3 x - 2} > 0$

Step 2.8.1.7.1.4

Multiply $- 2$ by $- 3$ .

$\frac{- 6 x^{2} + 4 x - 2 - 4 x + 6 x^{2} + 1 \cdot 2 + 1 (- 3 x)}{2 (2 - 3 x)} - \frac{6 x + 1}{3 x - 2} > 0$

Step 2.8.1.7.1.5

Multiply $2$ by $1$ .

$\frac{- 6 x^{2} + 4 x - 2 - 4 x + 6 x^{2} + 2 + 1 (- 3 x)}{2 (2 - 3 x)} - \frac{6 x + 1}{3 x - 2} > 0$

Step 2.8.1.7.1.6

Multiply $- 3 x$ by $1$ .

$\frac{- 6 x^{2} + 4 x - 2 - 4 x + 6 x^{2} + 2 - 3 x}{2 (2 - 3 x)} - \frac{6 x + 1}{3 x - 2} > 0$

Step 2.8.1.7.2

Subtract $3 x$ from $- 4 x$ .

$\frac{- 6 x^{2} + 4 x - 2 - 7 x + 6 x^{2} + 2}{2 (2 - 3 x)} - \frac{6 x + 1}{3 x - 2} > 0$

Step 2.8.1.8

Add $- 6 x^{2}$ and $6 x^{2}$ .

$\frac{4 x - 2 - 7 x + 0 + 2}{2 (2 - 3 x)} - \frac{6 x + 1}{3 x - 2} > 0$

Step 2.8.1.9

Subtract $7 x$ from $4 x$ .

$\frac{- 3 x - 2 + 0 + 2}{2 (2 - 3 x)} - \frac{6 x + 1}{3 x - 2} > 0$

Step 2.8.1.10

Add $- 3 x$ and $0$ .

$\frac{- 3 x - 2 + 2}{2 (2 - 3 x)} - \frac{6 x + 1}{3 x - 2} > 0$

Step 2.8.1.11

Add $- 2$ and $2$ .

$\frac{- 3 x + 0}{2 (2 - 3 x)} - \frac{6 x + 1}{3 x - 2} > 0$

Step 2.8.1.12

Add $- 3 x$ and $0$ .

$\frac{- 3 x}{2 (2 - 3 x)} - \frac{6 x + 1}{3 x - 2} > 0$

Step 2.8.2

Move the negative in front of the fraction.

$- \frac{3 x}{2 (2 - 3 x)} - \frac{6 x + 1}{3 x - 2} > 0$

Step 2.9

Factor $- 1$ out of $3 x$ .

$- \frac{3 x}{2 (2 - 3 x)} - \frac{6 x + 1}{- (- 3 x) - 2} > 0$

Step 2.10

Rewrite $- 2$ as $- 1 (2)$ .

$- \frac{3 x}{2 (2 - 3 x)} - \frac{6 x + 1}{- (- 3 x) - 1 (2)} > 0$

Step 2.11

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

$- \frac{3 x}{2 (2 - 3 x)} - \frac{6 x + 1}{- (- 3 x + 2)} > 0$

Step 2.12

Reorder terms.

$- \frac{3 x}{2 (- 3 x + 2)} - \frac{6 x + 1}{- (- 3 x + 2)} > 0$

Step 2.13

To write $- \frac{6 x + 1}{- (- 3 x + 2)}$ as a fraction with a common denominator, multiply by $\frac{- 2}{- 2}$ .

$- \frac{3 x}{2 (- 3 x + 2)} - \frac{6 x + 1}{- (- 3 x + 2)} \cdot \frac{- 2}{- 2} > 0$

Step 2.14

Write each expression with a common denominator of $2 (- 3 x + 2)$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.14.1

Multiply $\frac{6 x + 1}{- (- 3 x + 2)}$ by $\frac{- 2}{- 2}$ .

$- \frac{3 x}{2 (- 3 x + 2)} - \frac{(6 x + 1) \cdot - 2}{- (- 3 x + 2) \cdot - 2} > 0$

Step 2.14.2

Multiply $- 2$ by $- 1$ .

$- \frac{3 x}{2 (- 3 x + 2)} - \frac{(6 x + 1) \cdot - 2}{2 (- 3 x + 2)} > 0$

Step 2.15

Combine the numerators over the common denominator.

$\frac{- 3 x - (6 x + 1) \cdot - 2}{2 (- 3 x + 2)} > 0$

Step 2.16

Simplify the numerator.

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Step 2.16.1

Apply the distributive property.

$\frac{- 3 x + (- (6 x) - 1 \cdot 1) \cdot - 2}{2 (- 3 x + 2)} > 0$

Step 2.16.2

Multiply $6$ by $- 1$ .

$\frac{- 3 x + (- 6 x - 1 \cdot 1) \cdot - 2}{2 (- 3 x + 2)} > 0$

Step 2.16.3

Multiply $- 1$ by $1$ .

$\frac{- 3 x + (- 6 x - 1) \cdot - 2}{2 (- 3 x + 2)} > 0$

Step 2.16.4

Apply the distributive property.

$\frac{- 3 x - 6 x \cdot - 2 - 1 \cdot - 2}{2 (- 3 x + 2)} > 0$

Step 2.16.5

Multiply $- 2$ by $- 6$ .

$\frac{- 3 x + 12 x - 1 \cdot - 2}{2 (- 3 x + 2)} > 0$

Step 2.16.6

Multiply $- 1$ by $- 2$ .

$\frac{- 3 x + 12 x + 2}{2 (- 3 x + 2)} > 0$

Step 2.16.7

Add $- 3 x$ and $12 x$ .

$\frac{9 x + 2}{2 (- 3 x + 2)} > 0$

Step 2.17

Factor $- 1$ out of $- 3 x$ .

$\frac{9 x + 2}{2 (- (3 x) + 2)} > 0$

Step 2.18

Rewrite $2$ as $- 1 (- 2)$ .

$\frac{9 x + 2}{2 (- (3 x) - 1 (- 2))} > 0$

Step 2.19

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

$\frac{9 x + 2}{2 (- (3 x - 2))} > 0$

Step 2.20

Rewrite $- (3 x - 2)$ as $- 1 (3 x - 2)$ .

$\frac{9 x + 2}{2 (- 1 (3 x - 2))} > 0$

Step 2.21

Move the negative in front of the fraction.

$- \frac{9 x + 2}{2 (3 x - 2)} > 0$

Step 3

Find all the values where the expression switches from negative to positive by setting each factor equal to $0$ and solving.

$9 x + 2 = 0$

$3 x - 2 = 0$

Step 4

Subtract $2$ from both sides of the equation.

$9 x = - 2$

Step 5

Divide each term in $9 x = - 2$ by $9$ and simplify.

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Step 5.1

Divide each term in $9 x = - 2$ by $9$ .

$\frac{9 x}{9} = \frac{- 2}{9}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $9$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{9 x}{9} = \frac{- 2}{9}$

Step 5.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 2}{9}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Move the negative in front of the fraction.

$x = - \frac{2}{9}$

Step 6

Add $2$ to both sides of the equation.

$3 x = 2$

Step 7

Divide each term in $3 x = 2$ by $3$ and simplify.

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Step 7.1

Divide each term in $3 x = 2$ by $3$ .

$\frac{3 x}{3} = \frac{2}{3}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $3$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{2}{3}$

Step 7.2.1.2

Divide $x$ by $1$ .

$x = \frac{2}{3}$

Step 8

Solve for each factor to find the values where the absolute value expression goes from negative to positive.

$x = - \frac{2}{9}$

$x = \frac{2}{3}$

Step 9

Consolidate the solutions.

$x = - \frac{2}{9}, \frac{2}{3}$

Step 10

Find the domain of $- \frac{9 x + 2}{2 (3 x - 2)}$ .

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Step 10.1

Set the denominator in $\frac{9 x + 2}{2 (3 x - 2)}$ equal to $0$ to find where the expression is undefined.

$2 (3 x - 2) = 0$

Step 10.2

Solve for $x$ .

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Step 10.2.1

Divide each term in $2 (3 x - 2) = 0$ by $2$ and simplify.

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Step 10.2.1.1

Divide each term in $2 (3 x - 2) = 0$ by $2$ .

$\frac{2 (3 x - 2)}{2} = \frac{0}{2}$

Step 10.2.1.2

Simplify the left side.

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Step 10.2.1.2.1

Cancel the common factor of $2$ .

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Step 10.2.1.2.1.1

Cancel the common factor.

$\frac{2 (3 x - 2)}{2} = \frac{0}{2}$

Step 10.2.1.2.1.2

Divide $3 x - 2$ by $1$ .

$3 x - 2 = \frac{0}{2}$

Step 10.2.1.3

Simplify the right side.

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Step 10.2.1.3.1

Divide $0$ by $2$ .

$3 x - 2 = 0$

Step 10.2.2

Add $2$ to both sides of the equation.

$3 x = 2$

Step 10.2.3

Divide each term in $3 x = 2$ by $3$ and simplify.

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Step 10.2.3.1

Divide each term in $3 x = 2$ by $3$ .

$\frac{3 x}{3} = \frac{2}{3}$

Step 10.2.3.2

Simplify the left side.

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Step 10.2.3.2.1

Cancel the common factor of $3$ .

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Step 10.2.3.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{2}{3}$

Step 10.2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{2}{3}$

Step 10.3

The domain is all values of $x$ that make the expression defined.

$(- \infty, \frac{2}{3}) \cup (\frac{2}{3}, \infty)$

Step 11

Use each root to create test intervals.

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

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

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

Step 12

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 12.1

Test a value on the interval $x < - \frac{2}{9}$ to see if it makes the inequality true.

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Step 12.1.1

Choose a value on the interval $x < - \frac{2}{9}$ and see if this value makes the original inequality true.

$x = - 3$

Step 12.1.2

Replace $x$ with $- 3$ in the original inequality.

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

Step 12.1.3

The left side $- 3.\overline{09}$ is not greater than the right side $- 1.9\overline{54}$ , which means that the given statement is false.

False

Step 12.2

Test a value on the interval $- \frac{2}{9} < x < \frac{2}{3}$ to see if it makes the inequality true.

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Step 12.2.1

Choose a value on the interval $- \frac{2}{9} < x < \frac{2}{3}$ and see if this value makes the original inequality true.

$x = 0$

Step 12.2.2

Replace $x$ with $0$ in the original inequality.

$(0) - \frac{1}{2 - 3 \cdot 0} > \frac{2 (0) - 1}{2} + \frac{6 (0) + 1}{3 (0) - 2}$

Step 12.2.3

The left side $- 0.5$ is greater than the right side $- 1$ , which means that the given statement is always true.

True

Step 12.3

Test a value on the interval $x > \frac{2}{3}$ to see if it makes the inequality true.

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Step 12.3.1

Choose a value on the interval $x > \frac{2}{3}$ and see if this value makes the original inequality true.

$x = 3$

Step 12.3.2

Replace $x$ with $3$ in the original inequality.

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

Step 12.3.3

The left side $3.\overline{142857}$ is not greater than the right side $5.2\overline{142857}$ , which means that the given statement is false.

False

Step 12.4

Compare the intervals to determine which ones satisfy the original inequality.

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

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

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

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

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

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

Step 13

The solution consists of all of the true intervals.

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

Step 14

The result can be shown in multiple forms.

Inequality Form:

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

Interval Notation:

$(- \frac{2}{9}, \frac{2}{3})$

Step 15