Calculus Examples

$g (y) = \frac{y - 1}{y^{2} - 3 y + 3}$

Step 1

Rewrite the equation as $\frac{y - 1}{y^{2} - 3 y + 3} = x$ .

$\frac{y - 1}{y^{2} - 3 y + 3} = x$

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$y^{2} - 3 y + 3, 1$

Step 2.2

Remove parentheses.

$y^{2} - 3 y + 3, 1$

Step 2.3

The LCM of one and any expression is the expression.

$y^{2} - 3 y + 3$

Step 3

Multiply each term in $\frac{y - 1}{y^{2} - 3 y + 3} = x$ by $y^{2} - 3 y + 3$ to eliminate the fractions.

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Step 3.1

Multiply each term in $\frac{y - 1}{y^{2} - 3 y + 3} = x$ by $y^{2} - 3 y + 3$ .

$\frac{y - 1}{y^{2} - 3 y + 3} (y^{2} - 3 y + 3) = x (y^{2} - 3 y + 3)$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $y^{2} - 3 y + 3$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{y - 1}{y^{2} - 3 y + 3} (y^{2} - 3 y + 3) = x (y^{2} - 3 y + 3)$

Step 3.2.1.2

Rewrite the expression.

$y - 1 = x (y^{2} - 3 y + 3)$

Step 3.3

Simplify the right side.

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Step 3.3.1

Apply the distributive property.

$y - 1 = x y^{2} + x (- 3 y) + x \cdot 3$

Step 3.3.2

Simplify.

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Step 3.3.2.1

Rewrite using the commutative property of multiplication.

$y - 1 = x y^{2} - 3 x y + x \cdot 3$

Step 3.3.2.2

Move $3$ to the left of $x$ .

$y - 1 = x y^{2} - 3 x y + 3 \cdot x$

$y - 1 = x y^{2} - 3 x y + 3 x$

Step 4

Solve the equation.

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Step 4.1

Since $y$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$x y^{2} - 3 x y + 3 x = y - 1$

Step 4.2

Subtract $y$ from both sides of the equation.

$x y^{2} - 3 x y + 3 x - y = - 1$

Step 4.3

Add $1$ to both sides of the equation.

$x y^{2} - 3 x y + 3 x - y + 1 = 0$

Step 4.4

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4.5

Substitute the values $a = x$ , $b = - 3 x - 1$ , and $c = 3 x + 1$ into the quadratic formula and solve for $y$ .

$\frac{- (- 3 x - 1) \pm \sqrt{{(- 3 x - 1)}^{2} - 4 \cdot (x \cdot (3 x + 1))}}{2 x}$

Step 4.6

Simplify the numerator.

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Step 4.6.1

Apply the distributive property.

$y = \frac{- (- 3 x) + 1 \pm \sqrt{{(- 3 x - 1)}^{2} - 4 \cdot x \cdot (3 x + 1)}}{2 x}$

Step 4.6.2

Multiply $- 3$ by $- 1$ .

$y = \frac{3 x + 1 \pm \sqrt{{(- 3 x - 1)}^{2} - 4 \cdot x \cdot (3 x + 1)}}{2 x}$

Step 4.6.3

Multiply $- 1$ by $- 1$ .

$y = \frac{3 x + 1 \pm \sqrt{{(- 3 x - 1)}^{2} - 4 \cdot x \cdot (3 x + 1)}}{2 x}$

Step 4.6.4

Rewrite ${(- 3 x - 1)}^{2}$ as $(- 3 x - 1) (- 3 x - 1)$ .

$y = \frac{3 x + 1 \pm \sqrt{(- 3 x - 1) (- 3 x - 1) - 4 \cdot x \cdot (3 x + 1)}}{2 x}$

Step 4.6.5

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

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Step 4.6.5.1

Apply the distributive property.

$y = \frac{3 x + 1 \pm \sqrt{- 3 x (- 3 x - 1) - 1 (- 3 x - 1) - 4 \cdot x \cdot (3 x + 1)}}{2 x}$

Step 4.6.5.2

Apply the distributive property.

$y = \frac{3 x + 1 \pm \sqrt{- 3 x (- 3 x) - 3 x \cdot - 1 - 1 (- 3 x - 1) - 4 \cdot x \cdot (3 x + 1)}}{2 x}$

Step 4.6.5.3

Apply the distributive property.

$y = \frac{3 x + 1 \pm \sqrt{- 3 x (- 3 x) - 3 x \cdot - 1 - 1 (- 3 x) - 1 \cdot - 1 - 4 \cdot x \cdot (3 x + 1)}}{2 x}$

Step 4.6.6

Simplify and combine like terms.

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Step 4.6.6.1

Simplify each term.

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Step 4.6.6.1.1

Rewrite using the commutative property of multiplication.

$y = \frac{3 x + 1 \pm \sqrt{- 3 \cdot (- 3 x \cdot x) - 3 x \cdot - 1 - 1 (- 3 x) - 1 \cdot - 1 - 4 \cdot x \cdot (3 x + 1)}}{2 x}$

Step 4.6.6.1.2

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

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Step 4.6.6.1.2.1

Move $x$ .

$y = \frac{3 x + 1 \pm \sqrt{- 3 \cdot (- 3 (x \cdot x)) - 3 x \cdot - 1 - 1 (- 3 x) - 1 \cdot - 1 - 4 \cdot x \cdot (3 x + 1)}}{2 x}$

Step 4.6.6.1.2.2

Multiply $x$ by $x$ .

$y = \frac{3 x + 1 \pm \sqrt{- 3 \cdot (- 3 x^{2}) - 3 x \cdot - 1 - 1 (- 3 x) - 1 \cdot - 1 - 4 \cdot x \cdot (3 x + 1)}}{2 x}$

Step 4.6.6.1.3

Multiply $- 3$ by $- 3$ .

$y = \frac{3 x + 1 \pm \sqrt{9 x^{2} - 3 x \cdot - 1 - 1 (- 3 x) - 1 \cdot - 1 - 4 \cdot x \cdot (3 x + 1)}}{2 x}$

Step 4.6.6.1.4

Multiply $- 1$ by $- 3$ .

$y = \frac{3 x + 1 \pm \sqrt{9 x^{2} + 3 x - 1 (- 3 x) - 1 \cdot - 1 - 4 \cdot x \cdot (3 x + 1)}}{2 x}$

Step 4.6.6.1.5

Multiply $- 3$ by $- 1$ .

$y = \frac{3 x + 1 \pm \sqrt{9 x^{2} + 3 x + 3 x - 1 \cdot - 1 - 4 \cdot x \cdot (3 x + 1)}}{2 x}$

Step 4.6.6.1.6

Multiply $- 1$ by $- 1$ .

$y = \frac{3 x + 1 \pm \sqrt{9 x^{2} + 3 x + 3 x + 1 - 4 \cdot x \cdot (3 x + 1)}}{2 x}$

Step 4.6.6.2

Add $3 x$ and $3 x$ .

$y = \frac{3 x + 1 \pm \sqrt{9 x^{2} + 6 x + 1 - 4 \cdot x \cdot (3 x + 1)}}{2 x}$

Step 4.6.7

Apply the distributive property.

$y = \frac{3 x + 1 \pm \sqrt{9 x^{2} + 6 x + 1 - 4 x (3 x) - 4 x \cdot 1}}{2 x}$

Step 4.6.8

Rewrite using the commutative property of multiplication.

$y = \frac{3 x + 1 \pm \sqrt{9 x^{2} + 6 x + 1 - 4 \cdot (3 x \cdot x) - 4 x \cdot 1}}{2 x}$

Step 4.6.9

Multiply $- 4$ by $1$ .

$y = \frac{3 x + 1 \pm \sqrt{9 x^{2} + 6 x + 1 - 4 \cdot (3 x \cdot x) - 4 x}}{2 x}$

Step 4.6.10

Simplify each term.

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Step 4.6.10.1

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

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Step 4.6.10.1.1

Move $x$ .

$y = \frac{3 x + 1 \pm \sqrt{9 x^{2} + 6 x + 1 - 4 \cdot (3 (x \cdot x)) - 4 x}}{2 x}$

Step 4.6.10.1.2

Multiply $x$ by $x$ .

$y = \frac{3 x + 1 \pm \sqrt{9 x^{2} + 6 x + 1 - 4 \cdot (3 x^{2}) - 4 x}}{2 x}$

Step 4.6.10.2

Multiply $- 4$ by $3$ .

$y = \frac{3 x + 1 \pm \sqrt{9 x^{2} + 6 x + 1 - 12 x^{2} - 4 x}}{2 x}$

Step 4.6.11

Subtract $12 x^{2}$ from $9 x^{2}$ .

$y = \frac{3 x + 1 \pm \sqrt{- 3 x^{2} + 6 x + 1 - 4 x}}{2 x}$

Step 4.6.12

Subtract $4 x$ from $6 x$ .

$y = \frac{3 x + 1 \pm \sqrt{- 3 x^{2} + 2 x + 1}}{2 x}$

Step 4.6.13

Factor by grouping.

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Step 4.6.13.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 3 \cdot 1 = - 3$ and whose sum is $b = 2$ .

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Step 4.6.13.1.1

Factor $2$ out of $2 x$ .

$y = \frac{3 x + 1 \pm \sqrt{- 3 x^{2} + 2 (x) + 1}}{2 x}$

Step 4.6.13.1.2

Rewrite $2$ as $- 1$ plus $3$

$y = \frac{3 x + 1 \pm \sqrt{- 3 x^{2} + (- 1 + 3) x + 1}}{2 x}$

Step 4.6.13.1.3

Apply the distributive property.

$y = \frac{3 x + 1 \pm \sqrt{- 3 x^{2} - 1 x + 3 x + 1}}{2 x}$

Step 4.6.13.2

Factor out the greatest common factor from each group.

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Step 4.6.13.2.1

Group the first two terms and the last two terms.

$y = \frac{3 x + 1 \pm \sqrt{(- 3 x^{2} - 1 x) + 3 x + 1}}{2 x}$

Step 4.6.13.2.2

Factor out the greatest common factor (GCF) from each group.

$y = \frac{3 x + 1 \pm \sqrt{x (- 3 x - 1) - (- 3 x - 1)}}{2 x}$

Step 4.6.13.3

Factor the polynomial by factoring out the greatest common factor, $- 3 x - 1$ .

$y = \frac{3 x + 1 \pm \sqrt{(- 3 x - 1) (x - 1)}}{2 x}$

Step 4.7

Change the $\pm$ to $+$ .

$y = \frac{3 x + 1 + \sqrt{(- 3 x - 1) (x - 1)}}{2 x}$

Step 4.8

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 4.8.1

Simplify the numerator.

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Step 4.8.1.1

Apply the distributive property.

$y = \frac{- (- 3 x) + 1 \pm \sqrt{{(- 3 x - 1)}^{2} - 4 \cdot x \cdot (3 x + 1)}}{2 x}$

Step 4.8.1.2

Multiply $- 3$ by $- 1$ .

$y = \frac{3 x + 1 \pm \sqrt{{(- 3 x - 1)}^{2} - 4 \cdot x \cdot (3 x + 1)}}{2 x}$

Step 4.8.1.3

Multiply $- 1$ by $- 1$ .

$y = \frac{3 x + 1 \pm \sqrt{{(- 3 x - 1)}^{2} - 4 \cdot x \cdot (3 x + 1)}}{2 x}$

Step 4.8.1.4

Rewrite ${(- 3 x - 1)}^{2}$ as $(- 3 x - 1) (- 3 x - 1)$ .

$y = \frac{3 x + 1 \pm \sqrt{(- 3 x - 1) (- 3 x - 1) - 4 \cdot x \cdot (3 x + 1)}}{2 x}$

Step 4.8.1.5

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

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Step 4.8.1.5.1

Apply the distributive property.

$y = \frac{3 x + 1 \pm \sqrt{- 3 x (- 3 x - 1) - 1 (- 3 x - 1) - 4 \cdot x \cdot (3 x + 1)}}{2 x}$

Step 4.8.1.5.2

Apply the distributive property.

$y = \frac{3 x + 1 \pm \sqrt{- 3 x (- 3 x) - 3 x \cdot - 1 - 1 (- 3 x - 1) - 4 \cdot x \cdot (3 x + 1)}}{2 x}$

Step 4.8.1.5.3

Apply the distributive property.

$y = \frac{3 x + 1 \pm \sqrt{- 3 x (- 3 x) - 3 x \cdot - 1 - 1 (- 3 x) - 1 \cdot - 1 - 4 \cdot x \cdot (3 x + 1)}}{2 x}$

Step 4.8.1.6

Simplify and combine like terms.

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Step 4.8.1.6.1

Simplify each term.

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Step 4.8.1.6.1.1

Rewrite using the commutative property of multiplication.

$y = \frac{3 x + 1 \pm \sqrt{- 3 \cdot (- 3 x \cdot x) - 3 x \cdot - 1 - 1 (- 3 x) - 1 \cdot - 1 - 4 \cdot x \cdot (3 x + 1)}}{2 x}$

Step 4.8.1.6.1.2

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

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Step 4.8.1.6.1.2.1

Move $x$ .

$y = \frac{3 x + 1 \pm \sqrt{- 3 \cdot (- 3 (x \cdot x)) - 3 x \cdot - 1 - 1 (- 3 x) - 1 \cdot - 1 - 4 \cdot x \cdot (3 x + 1)}}{2 x}$

Step 4.8.1.6.1.2.2

Multiply $x$ by $x$ .

$y = \frac{3 x + 1 \pm \sqrt{- 3 \cdot (- 3 x^{2}) - 3 x \cdot - 1 - 1 (- 3 x) - 1 \cdot - 1 - 4 \cdot x \cdot (3 x + 1)}}{2 x}$

Step 4.8.1.6.1.3

Multiply $- 3$ by $- 3$ .

$y = \frac{3 x + 1 \pm \sqrt{9 x^{2} - 3 x \cdot - 1 - 1 (- 3 x) - 1 \cdot - 1 - 4 \cdot x \cdot (3 x + 1)}}{2 x}$

Step 4.8.1.6.1.4

Multiply $- 1$ by $- 3$ .

$y = \frac{3 x + 1 \pm \sqrt{9 x^{2} + 3 x - 1 (- 3 x) - 1 \cdot - 1 - 4 \cdot x \cdot (3 x + 1)}}{2 x}$

Step 4.8.1.6.1.5

Multiply $- 3$ by $- 1$ .

$y = \frac{3 x + 1 \pm \sqrt{9 x^{2} + 3 x + 3 x - 1 \cdot - 1 - 4 \cdot x \cdot (3 x + 1)}}{2 x}$

Step 4.8.1.6.1.6

Multiply $- 1$ by $- 1$ .

$y = \frac{3 x + 1 \pm \sqrt{9 x^{2} + 3 x + 3 x + 1 - 4 \cdot x \cdot (3 x + 1)}}{2 x}$

Step 4.8.1.6.2

Add $3 x$ and $3 x$ .

$y = \frac{3 x + 1 \pm \sqrt{9 x^{2} + 6 x + 1 - 4 \cdot x \cdot (3 x + 1)}}{2 x}$

Step 4.8.1.7

Apply the distributive property.

$y = \frac{3 x + 1 \pm \sqrt{9 x^{2} + 6 x + 1 - 4 x (3 x) - 4 x \cdot 1}}{2 x}$

Step 4.8.1.8

Rewrite using the commutative property of multiplication.

$y = \frac{3 x + 1 \pm \sqrt{9 x^{2} + 6 x + 1 - 4 \cdot (3 x \cdot x) - 4 x \cdot 1}}{2 x}$

Step 4.8.1.9

Multiply $- 4$ by $1$ .

$y = \frac{3 x + 1 \pm \sqrt{9 x^{2} + 6 x + 1 - 4 \cdot (3 x \cdot x) - 4 x}}{2 x}$

Step 4.8.1.10

Simplify each term.

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Step 4.8.1.10.1

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

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Step 4.8.1.10.1.1

Move $x$ .

$y = \frac{3 x + 1 \pm \sqrt{9 x^{2} + 6 x + 1 - 4 \cdot (3 (x \cdot x)) - 4 x}}{2 x}$

Step 4.8.1.10.1.2

Multiply $x$ by $x$ .

$y = \frac{3 x + 1 \pm \sqrt{9 x^{2} + 6 x + 1 - 4 \cdot (3 x^{2}) - 4 x}}{2 x}$

Step 4.8.1.10.2

Multiply $- 4$ by $3$ .

$y = \frac{3 x + 1 \pm \sqrt{9 x^{2} + 6 x + 1 - 12 x^{2} - 4 x}}{2 x}$

Step 4.8.1.11

Subtract $12 x^{2}$ from $9 x^{2}$ .

$y = \frac{3 x + 1 \pm \sqrt{- 3 x^{2} + 6 x + 1 - 4 x}}{2 x}$

Step 4.8.1.12

Subtract $4 x$ from $6 x$ .

$y = \frac{3 x + 1 \pm \sqrt{- 3 x^{2} + 2 x + 1}}{2 x}$

Step 4.8.1.13

Factor by grouping.

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Step 4.8.1.13.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 3 \cdot 1 = - 3$ and whose sum is $b = 2$ .

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Step 4.8.1.13.1.1

Factor $2$ out of $2 x$ .

$y = \frac{3 x + 1 \pm \sqrt{- 3 x^{2} + 2 (x) + 1}}{2 x}$

Step 4.8.1.13.1.2

Rewrite $2$ as $- 1$ plus $3$

$y = \frac{3 x + 1 \pm \sqrt{- 3 x^{2} + (- 1 + 3) x + 1}}{2 x}$

Step 4.8.1.13.1.3

Apply the distributive property.

$y = \frac{3 x + 1 \pm \sqrt{- 3 x^{2} - 1 x + 3 x + 1}}{2 x}$

Step 4.8.1.13.2

Factor out the greatest common factor from each group.

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Step 4.8.1.13.2.1

Group the first two terms and the last two terms.

$y = \frac{3 x + 1 \pm \sqrt{(- 3 x^{2} - 1 x) + 3 x + 1}}{2 x}$

Step 4.8.1.13.2.2

Factor out the greatest common factor (GCF) from each group.

$y = \frac{3 x + 1 \pm \sqrt{x (- 3 x - 1) - (- 3 x - 1)}}{2 x}$

Step 4.8.1.13.3

Factor the polynomial by factoring out the greatest common factor, $- 3 x - 1$ .

$y = \frac{3 x + 1 \pm \sqrt{(- 3 x - 1) (x - 1)}}{2 x}$

Step 4.8.2

Change the $\pm$ to $-$ .

$y = \frac{3 x + 1 - \sqrt{(- 3 x - 1) (x - 1)}}{2 x}$

Step 4.9

The final answer is the combination of both solutions.

$y = \frac{3 x + 1 + \sqrt{(- 3 x - 1) (x - 1)}}{2 x}$

$y = \frac{3 x + 1 - \sqrt{(- 3 x - 1) (x - 1)}}{2 x}$

$y = \frac{3 x + 1 + \sqrt{(- 3 x - 1) (x - 1)}}{2 x}$

$y = \frac{3 x + 1 - \sqrt{(- 3 x - 1) (x - 1)}}{2 x}$

Step 5

Set the radicand in $\sqrt{(- 3 x - 1) (x - 1)}$ greater than or equal to $0$ to find where the expression is defined.

$(- 3 x - 1) (x - 1) \geq 0$

Step 6

Solve for $x$ .

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Step 6.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$- 3 x - 1 = 0$

$x - 1 = 0$

Step 6.2

Set $- 3 x - 1$ equal to $0$ and solve for $x$ .

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Step 6.2.1

Set $- 3 x - 1$ equal to $0$ .

$- 3 x - 1 = 0$

Step 6.2.2

Solve $- 3 x - 1 = 0$ for $x$ .

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Step 6.2.2.1

Add $1$ to both sides of the equation.

$- 3 x = 1$

Step 6.2.2.2

Divide each term in $- 3 x = 1$ by $- 3$ and simplify.

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Step 6.2.2.2.1

Divide each term in $- 3 x = 1$ by $- 3$ .

$\frac{- 3 x}{- 3} = \frac{1}{- 3}$

Step 6.2.2.2.2

Simplify the left side.

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Step 6.2.2.2.2.1

Cancel the common factor of $- 3$ .

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Step 6.2.2.2.2.1.1

Cancel the common factor.

$\frac{- 3 x}{- 3} = \frac{1}{- 3}$

Step 6.2.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{- 3}$

Step 6.2.2.2.3

Simplify the right side.

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Step 6.2.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{1}{3}$

Step 6.3

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 6.3.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 6.3.2

Add $1$ to both sides of the equation.

$x = 1$

Step 6.4

The final solution is all the values that make $(- 3 x - 1) (x - 1) \geq 0$ true.

$x = - \frac{1}{3}, 1$

Step 6.5

Use each root to create test intervals.

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

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

$x > 1$

Step 6.6

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 6.6.1

Test a value on the interval $x < - \frac{1}{3}$ to see if it makes the inequality true.

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Step 6.6.1.1

Choose a value on the interval $x < - \frac{1}{3}$ and see if this value makes the original inequality true.

$x = - 3$

Step 6.6.1.2

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

$(- 3 \cdot - 3 - 1) ((- 3) - 1) \geq 0$

Step 6.6.1.3

The left side $- 32$ is less than the right side $0$ , which means that the given statement is false.

False

Step 6.6.2

Test a value on the interval $- \frac{1}{3} < x < 1$ to see if it makes the inequality true.

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Step 6.6.2.1

Choose a value on the interval $- \frac{1}{3} < x < 1$ and see if this value makes the original inequality true.

$x = 0$

Step 6.6.2.2

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

$(- 3 \cdot 0 - 1) ((0) - 1) \geq 0$

Step 6.6.2.3

The left side $1$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 6.6.3

Test a value on the interval $x > 1$ to see if it makes the inequality true.

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Step 6.6.3.1

Choose a value on the interval $x > 1$ and see if this value makes the original inequality true.

$x = 4$

Step 6.6.3.2

Replace $x$ with $4$ in the original inequality.

$(- 3 \cdot 4 - 1) ((4) - 1) \geq 0$

Step 6.6.3.3

The left side $- 39$ is less than the right side $0$ , which means that the given statement is false.

False

Step 6.6.4

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

$x < - \frac{1}{3}$ False

$- \frac{1}{3} < x < 1$ True

$x > 1$ False

$x < - \frac{1}{3}$ False

$- \frac{1}{3} < x < 1$ True

$x > 1$ False

Step 6.7

The solution consists of all of the true intervals.

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

Step 7

Set the denominator in $\frac{3 x + 1 + \sqrt{(- 3 x - 1) (x - 1)}}{2 x}$ equal to $0$ to find where the expression is undefined.

$2 x = 0$

Step 8

Divide each term in $2 x = 0$ by $2$ and simplify.

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Step 8.1

Divide each term in $2 x = 0$ by $2$ .

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

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $2$ .

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Step 8.2.1.1

Cancel the common factor.

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

Step 8.2.1.2

Divide $x$ by $1$ .

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

Step 8.3

Simplify the right side.

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Step 8.3.1

Divide $0$ by $2$ .

$x = 0$

Step 9

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

Interval Notation:

$[- \frac{1}{3}, 0) \cup (0, 1]$

Set-Builder Notation:

${x | - \frac{1}{3} \leq x \leq 1, x \neq 0}$

Step 10

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${y | y \in ℝ}$

Step 11

Determine the domain and range.

Domain: $[- \frac{1}{3}, 0) \cup (0, 1], {x | - \frac{1}{3} \leq x \leq 1, x \neq 0}$

Range: $(- \infty, \infty), {y | y \in ℝ}$

Step 12