Linear Algebra Examples

$3 x y + 2 (x^{2} + y^{2}) x = 7$

Step 1

Simplify each term.

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Step 1.1

Apply the distributive property.

$3 x y + (2 x^{2} + 2 y^{2}) x = 7$

Step 1.2

Apply the distributive property.

$3 x y + 2 x^{2} x + 2 y^{2} x = 7$

Step 1.3

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 1.3.1

Move $x$ .

$3 x y + 2 (x \cdot x^{2}) + 2 y^{2} x = 7$

Step 1.3.2

Multiply $x$ by $x^{2}$ .

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Step 1.3.2.1

Raise $x$ to the power of $1$ .

$3 x y + 2 (x^{1} x^{2}) + 2 y^{2} x = 7$

Step 1.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$3 x y + 2 x^{1 + 2} + 2 y^{2} x = 7$

Step 1.3.3

Add $1$ and $2$ .

$3 x y + 2 x^{3} + 2 y^{2} x = 7$

Step 2

Subtract $7$ from both sides of the equation.

$3 x y + 2 x^{3} + 2 y^{2} x - 7 = 0$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = 2 x$ , $b = 3 x$ , and $c = 2 x^{3} - 7$ into the quadratic formula and solve for $y$ .

$\frac{- 3 x \pm \sqrt{{(3 x)}^{2} - 4 \cdot (2 x \cdot (2 x^{3} - 7))}}{2 (2 x)}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Apply the product rule to $3 x$ .

$y = \frac{- 3 x \pm \sqrt{3^{2} x^{2} - 4 \cdot (2 x) \cdot (2 x^{3} - 7)}}{2 \cdot (2 x)}$

Step 5.1.2

Raise $3$ to the power of $2$ .

$y = \frac{- 3 x \pm \sqrt{9 x^{2} - 4 \cdot (2 x) \cdot (2 x^{3} - 7)}}{2 \cdot (2 x)}$

Step 5.1.3

Multiply $- 4$ by $2$ .

$y = \frac{- 3 x \pm \sqrt{9 x^{2} - 8 x \cdot (2 x^{3} - 7)}}{2 \cdot (2 x)}$

Step 5.1.4

Apply the distributive property.

$y = \frac{- 3 x \pm \sqrt{9 x^{2} - 8 x (2 x^{3}) - 8 x \cdot - 7}}{2 \cdot (2 x)}$

Step 5.1.5

Rewrite using the commutative property of multiplication.

$y = \frac{- 3 x \pm \sqrt{9 x^{2} - 8 \cdot (2 x \cdot x^{3}) - 8 x \cdot - 7}}{2 \cdot (2 x)}$

Step 5.1.6

Multiply $- 7$ by $- 8$ .

$y = \frac{- 3 x \pm \sqrt{9 x^{2} - 8 \cdot (2 x \cdot x^{3}) + 56 x}}{2 \cdot (2 x)}$

Step 5.1.7

Simplify each term.

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Step 5.1.7.1

Multiply $x$ by $x^{3}$ by adding the exponents.

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Step 5.1.7.1.1

Move $x^{3}$ .

$y = \frac{- 3 x \pm \sqrt{9 x^{2} - 8 \cdot (2 (x^{3} x)) + 56 x}}{2 \cdot (2 x)}$

Step 5.1.7.1.2

Multiply $x^{3}$ by $x$ .

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Step 5.1.7.1.2.1

Raise $x$ to the power of $1$ .

$y = \frac{- 3 x \pm \sqrt{9 x^{2} - 8 \cdot (2 (x^{3} x)) + 56 x}}{2 \cdot (2 x)}$

Step 5.1.7.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \frac{- 3 x \pm \sqrt{9 x^{2} - 8 \cdot (2 x^{3 + 1}) + 56 x}}{2 \cdot (2 x)}$

Step 5.1.7.1.3

Add $3$ and $1$ .

$y = \frac{- 3 x \pm \sqrt{9 x^{2} - 8 \cdot (2 x^{4}) + 56 x}}{2 \cdot (2 x)}$

Step 5.1.7.2

Multiply $- 8$ by $2$ .

$y = \frac{- 3 x \pm \sqrt{9 x^{2} - 16 x^{4} + 56 x}}{2 \cdot (2 x)}$

Step 5.1.8

Reorder terms.

$y = \frac{- 3 x \pm \sqrt{- 16 x^{4} + 9 x^{2} + 56 x}}{2 \cdot (2 x)}$

Step 5.1.9

Factor $x$ out of $- 16 x^{4} + 9 x^{2} + 56 x$ .

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Step 5.1.9.1

Factor $x$ out of $- 16 x^{4}$ .

$y = \frac{- 3 x \pm \sqrt{x (- 16 x^{3}) + 9 x^{2} + 56 x}}{2 \cdot (2 x)}$

Step 5.1.9.2

Factor $x$ out of $9 x^{2}$ .

$y = \frac{- 3 x \pm \sqrt{x (- 16 x^{3}) + x (9 x) + 56 x}}{2 \cdot (2 x)}$

Step 5.1.9.3

Factor $x$ out of $56 x$ .

$y = \frac{- 3 x \pm \sqrt{x (- 16 x^{3}) + x (9 x) + x \cdot 56}}{2 \cdot (2 x)}$

Step 5.1.9.4

Factor $x$ out of $x (- 16 x^{3}) + x (9 x)$ .

$y = \frac{- 3 x \pm \sqrt{x (- 16 x^{3} + 9 x) + x \cdot 56}}{2 \cdot (2 x)}$

Step 5.1.9.5

Factor $x$ out of $x (- 16 x^{3} + 9 x) + x \cdot 56$ .

$y = \frac{- 3 x \pm \sqrt{x (- 16 x^{3} + 9 x + 56)}}{2 \cdot (2 x)}$

Step 5.2

Multiply $2$ by $2$ .

$y = \frac{- 3 x \pm \sqrt{x (- 16 x^{3} + 9 x + 56)}}{4 x}$

Step 6

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

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Apply the product rule to $3 x$ .

$y = \frac{- 3 x \pm \sqrt{3^{2} x^{2} - 4 \cdot (2 x) \cdot (2 x^{3} - 7)}}{2 \cdot (2 x)}$

Step 6.1.2

Raise $3$ to the power of $2$ .

$y = \frac{- 3 x \pm \sqrt{9 x^{2} - 4 \cdot (2 x) \cdot (2 x^{3} - 7)}}{2 \cdot (2 x)}$

Step 6.1.3

Multiply $- 4$ by $2$ .

$y = \frac{- 3 x \pm \sqrt{9 x^{2} - 8 x \cdot (2 x^{3} - 7)}}{2 \cdot (2 x)}$

Step 6.1.4

Apply the distributive property.

$y = \frac{- 3 x \pm \sqrt{9 x^{2} - 8 x (2 x^{3}) - 8 x \cdot - 7}}{2 \cdot (2 x)}$

Step 6.1.5

Rewrite using the commutative property of multiplication.

$y = \frac{- 3 x \pm \sqrt{9 x^{2} - 8 \cdot (2 x \cdot x^{3}) - 8 x \cdot - 7}}{2 \cdot (2 x)}$

Step 6.1.6

Multiply $- 7$ by $- 8$ .

$y = \frac{- 3 x \pm \sqrt{9 x^{2} - 8 \cdot (2 x \cdot x^{3}) + 56 x}}{2 \cdot (2 x)}$

Step 6.1.7

Simplify each term.

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Step 6.1.7.1

Multiply $x$ by $x^{3}$ by adding the exponents.

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Step 6.1.7.1.1

Move $x^{3}$ .

$y = \frac{- 3 x \pm \sqrt{9 x^{2} - 8 \cdot (2 (x^{3} x)) + 56 x}}{2 \cdot (2 x)}$

Step 6.1.7.1.2

Multiply $x^{3}$ by $x$ .

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Step 6.1.7.1.2.1

Raise $x$ to the power of $1$ .

$y = \frac{- 3 x \pm \sqrt{9 x^{2} - 8 \cdot (2 (x^{3} x)) + 56 x}}{2 \cdot (2 x)}$

Step 6.1.7.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \frac{- 3 x \pm \sqrt{9 x^{2} - 8 \cdot (2 x^{3 + 1}) + 56 x}}{2 \cdot (2 x)}$

Step 6.1.7.1.3

Add $3$ and $1$ .

$y = \frac{- 3 x \pm \sqrt{9 x^{2} - 8 \cdot (2 x^{4}) + 56 x}}{2 \cdot (2 x)}$

Step 6.1.7.2

Multiply $- 8$ by $2$ .

$y = \frac{- 3 x \pm \sqrt{9 x^{2} - 16 x^{4} + 56 x}}{2 \cdot (2 x)}$

Step 6.1.8

Reorder terms.

$y = \frac{- 3 x \pm \sqrt{- 16 x^{4} + 9 x^{2} + 56 x}}{2 \cdot (2 x)}$

Step 6.1.9

Factor $x$ out of $- 16 x^{4} + 9 x^{2} + 56 x$ .

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Step 6.1.9.1

Factor $x$ out of $- 16 x^{4}$ .

$y = \frac{- 3 x \pm \sqrt{x (- 16 x^{3}) + 9 x^{2} + 56 x}}{2 \cdot (2 x)}$

Step 6.1.9.2

Factor $x$ out of $9 x^{2}$ .

$y = \frac{- 3 x \pm \sqrt{x (- 16 x^{3}) + x (9 x) + 56 x}}{2 \cdot (2 x)}$

Step 6.1.9.3

Factor $x$ out of $56 x$ .

$y = \frac{- 3 x \pm \sqrt{x (- 16 x^{3}) + x (9 x) + x \cdot 56}}{2 \cdot (2 x)}$

Step 6.1.9.4

Factor $x$ out of $x (- 16 x^{3}) + x (9 x)$ .

$y = \frac{- 3 x \pm \sqrt{x (- 16 x^{3} + 9 x) + x \cdot 56}}{2 \cdot (2 x)}$

Step 6.1.9.5

Factor $x$ out of $x (- 16 x^{3} + 9 x) + x \cdot 56$ .

$y = \frac{- 3 x \pm \sqrt{x (- 16 x^{3} + 9 x + 56)}}{2 \cdot (2 x)}$

Step 6.2

Multiply $2$ by $2$ .

$y = \frac{- 3 x \pm \sqrt{x (- 16 x^{3} + 9 x + 56)}}{4 x}$

Step 6.3

Change the $\pm$ to $+$ .

$y = \frac{- 3 x + \sqrt{x (- 16 x^{3} + 9 x + 56)}}{4 x}$

Step 6.4

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

$y = \frac{- (3 x) + \sqrt{x (- 16 x^{3} + 9 x + 56)}}{4 x}$

Step 6.5

Factor $- 1$ out of $\sqrt{x (- 16 x^{3} + 9 x + 56)}$ .

$y = \frac{- (3 x) - 1 (- \sqrt{x (- 16 x^{3} + 9 x + 56)})}{4 x}$

Step 6.6

Factor $- 1$ out of $- (3 x) - 1 (- \sqrt{x (- 16 x^{3} + 9 x + 56)})$ .

$y = \frac{- (3 x - \sqrt{x (- 16 x^{3} + 9 x + 56)})}{4 x}$

Step 6.7

Rewrite $- (3 x - \sqrt{x (- 16 x^{3} + 9 x + 56)})$ as $- 1 (3 x - \sqrt{x (- 16 x^{3} + 9 x + 56)})$ .

$y = \frac{- 1 (3 x - \sqrt{x (- 16 x^{3} + 9 x + 56)})}{4 x}$

Step 6.8

Move the negative in front of the fraction.

$y = - \frac{3 x - \sqrt{x (- 16 x^{3} + 9 x + 56)}}{4 x}$

Step 7

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

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Apply the product rule to $3 x$ .

$y = \frac{- 3 x \pm \sqrt{3^{2} x^{2} - 4 \cdot (2 x) \cdot (2 x^{3} - 7)}}{2 \cdot (2 x)}$

Step 7.1.2

Raise $3$ to the power of $2$ .

$y = \frac{- 3 x \pm \sqrt{9 x^{2} - 4 \cdot (2 x) \cdot (2 x^{3} - 7)}}{2 \cdot (2 x)}$

Step 7.1.3

Multiply $- 4$ by $2$ .

$y = \frac{- 3 x \pm \sqrt{9 x^{2} - 8 x \cdot (2 x^{3} - 7)}}{2 \cdot (2 x)}$

Step 7.1.4

Apply the distributive property.

$y = \frac{- 3 x \pm \sqrt{9 x^{2} - 8 x (2 x^{3}) - 8 x \cdot - 7}}{2 \cdot (2 x)}$

Step 7.1.5

Rewrite using the commutative property of multiplication.

$y = \frac{- 3 x \pm \sqrt{9 x^{2} - 8 \cdot (2 x \cdot x^{3}) - 8 x \cdot - 7}}{2 \cdot (2 x)}$

Step 7.1.6

Multiply $- 7$ by $- 8$ .

$y = \frac{- 3 x \pm \sqrt{9 x^{2} - 8 \cdot (2 x \cdot x^{3}) + 56 x}}{2 \cdot (2 x)}$

Step 7.1.7

Simplify each term.

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Step 7.1.7.1

Multiply $x$ by $x^{3}$ by adding the exponents.

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Step 7.1.7.1.1

Move $x^{3}$ .

$y = \frac{- 3 x \pm \sqrt{9 x^{2} - 8 \cdot (2 (x^{3} x)) + 56 x}}{2 \cdot (2 x)}$

Step 7.1.7.1.2

Multiply $x^{3}$ by $x$ .

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Step 7.1.7.1.2.1

Raise $x$ to the power of $1$ .

$y = \frac{- 3 x \pm \sqrt{9 x^{2} - 8 \cdot (2 (x^{3} x)) + 56 x}}{2 \cdot (2 x)}$

Step 7.1.7.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \frac{- 3 x \pm \sqrt{9 x^{2} - 8 \cdot (2 x^{3 + 1}) + 56 x}}{2 \cdot (2 x)}$

Step 7.1.7.1.3

Add $3$ and $1$ .

$y = \frac{- 3 x \pm \sqrt{9 x^{2} - 8 \cdot (2 x^{4}) + 56 x}}{2 \cdot (2 x)}$

Step 7.1.7.2

Multiply $- 8$ by $2$ .

$y = \frac{- 3 x \pm \sqrt{9 x^{2} - 16 x^{4} + 56 x}}{2 \cdot (2 x)}$

Step 7.1.8

Reorder terms.

$y = \frac{- 3 x \pm \sqrt{- 16 x^{4} + 9 x^{2} + 56 x}}{2 \cdot (2 x)}$

Step 7.1.9

Factor $x$ out of $- 16 x^{4} + 9 x^{2} + 56 x$ .

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Step 7.1.9.1

Factor $x$ out of $- 16 x^{4}$ .

$y = \frac{- 3 x \pm \sqrt{x (- 16 x^{3}) + 9 x^{2} + 56 x}}{2 \cdot (2 x)}$

Step 7.1.9.2

Factor $x$ out of $9 x^{2}$ .

$y = \frac{- 3 x \pm \sqrt{x (- 16 x^{3}) + x (9 x) + 56 x}}{2 \cdot (2 x)}$

Step 7.1.9.3

Factor $x$ out of $56 x$ .

$y = \frac{- 3 x \pm \sqrt{x (- 16 x^{3}) + x (9 x) + x \cdot 56}}{2 \cdot (2 x)}$

Step 7.1.9.4

Factor $x$ out of $x (- 16 x^{3}) + x (9 x)$ .

$y = \frac{- 3 x \pm \sqrt{x (- 16 x^{3} + 9 x) + x \cdot 56}}{2 \cdot (2 x)}$

Step 7.1.9.5

Factor $x$ out of $x (- 16 x^{3} + 9 x) + x \cdot 56$ .

$y = \frac{- 3 x \pm \sqrt{x (- 16 x^{3} + 9 x + 56)}}{2 \cdot (2 x)}$

Step 7.2

Multiply $2$ by $2$ .

$y = \frac{- 3 x \pm \sqrt{x (- 16 x^{3} + 9 x + 56)}}{4 x}$

Step 7.3

Change the $\pm$ to $-$ .

$y = \frac{- 3 x - \sqrt{x (- 16 x^{3} + 9 x + 56)}}{4 x}$

Step 7.4

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

$y = \frac{- (3 x) - \sqrt{x (- 16 x^{3} + 9 x + 56)}}{4 x}$

Step 7.5

Factor $- 1$ out of $- \sqrt{x (- 16 x^{3} + 9 x + 56)}$ .

$y = \frac{- (3 x) - (\sqrt{x (- 16 x^{3} + 9 x + 56)})}{4 x}$

Step 7.6

Factor $- 1$ out of $- (3 x) - (\sqrt{x (- 16 x^{3} + 9 x + 56)})$ .

$y = \frac{- (3 x + \sqrt{x (- 16 x^{3} + 9 x + 56)})}{4 x}$

Step 7.7

Rewrite $- (3 x + \sqrt{x (- 16 x^{3} + 9 x + 56)})$ as $- 1 (3 x + \sqrt{x (- 16 x^{3} + 9 x + 56)})$ .

$y = \frac{- 1 (3 x + \sqrt{x (- 16 x^{3} + 9 x + 56)})}{4 x}$

Step 7.8

Move the negative in front of the fraction.

$y = - \frac{3 x + \sqrt{x (- 16 x^{3} + 9 x + 56)}}{4 x}$

Step 8

The final answer is the combination of both solutions.

$y = - \frac{3 x - \sqrt{x (- 16 x^{3} + 9 x + 56)}}{4 x}$

$y = - \frac{3 x + \sqrt{x (- 16 x^{3} + 9 x + 56)}}{4 x}$

Step 9

Set the radicand in $\sqrt{x (- 16 x^{3} + 9 x + 56)}$ greater than or equal to $0$ to find where the expression is defined.

$x (- 16 x^{3} + 9 x + 56) \geq 0$

Step 10

Solve for $x$ .

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Step 10.1

Simplify $x (- 16 x^{3} + 9 x + 56)$ .

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Step 10.1.1

Apply the distributive property.

$x (- 16 x^{3}) + x (9 x) + x \cdot 56 \geq 0$

Step 10.1.2

Simplify.

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Step 10.1.2.1

Rewrite using the commutative property of multiplication.

$- 16 x \cdot x^{3} + x (9 x) + x \cdot 56 \geq 0$

Step 10.1.2.2

Rewrite using the commutative property of multiplication.

$- 16 x \cdot x^{3} + 9 x \cdot x + x \cdot 56 \geq 0$

Step 10.1.2.3

Move $56$ to the left of $x$ .

$- 16 x \cdot x^{3} + 9 x \cdot x + 56 \cdot x \geq 0$

Step 10.1.3

Simplify each term.

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Step 10.1.3.1

Multiply $x$ by $x^{3}$ by adding the exponents.

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Step 10.1.3.1.1

Move $x^{3}$ .

$- 16 (x^{3} x) + 9 x \cdot x + 56 \cdot x \geq 0$

Step 10.1.3.1.2

Multiply $x^{3}$ by $x$ .

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Step 10.1.3.1.2.1

Raise $x$ to the power of $1$ .

$- 16 (x^{3} x^{1}) + 9 x \cdot x + 56 \cdot x \geq 0$

Step 10.1.3.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 16 x^{3 + 1} + 9 x \cdot x + 56 \cdot x \geq 0$

Step 10.1.3.1.3

Add $3$ and $1$ .

$- 16 x^{4} + 9 x \cdot x + 56 \cdot x \geq 0$

Step 10.1.3.2

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

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Step 10.1.3.2.1

Move $x$ .

$- 16 x^{4} + 9 (x \cdot x) + 56 \cdot x \geq 0$

Step 10.1.3.2.2

Multiply $x$ by $x$ .

$- 16 x^{4} + 9 x^{2} + 56 \cdot x \geq 0$

$- 16 x^{4} + 9 x^{2} + 56 x \geq 0$

Step 10.2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx 0, 1.64153795$

Step 10.3

Use each root to create test intervals.

$x < 0$

$0 < x < 1.64153795$

$x > 1.64153795$

Step 10.4

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 10.4.1

Test a value on the interval $x < 0$ to see if it makes the inequality true.

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Step 10.4.1.1

Choose a value on the interval $x < 0$ and see if this value makes the original inequality true.

$x = - 2$

Step 10.4.1.2

Replace $x$ with $- 2$ in the original inequality.

$(- 2) (- 16 {(- 2)}^{3} + 9 (- 2) + 56) \geq 0$

Step 10.4.1.3

The left side $- 332$ is less than the right side $0$ , which means that the given statement is false.

False

Step 10.4.2

Test a value on the interval $0 < x < 1.64153795$ to see if it makes the inequality true.

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Step 10.4.2.1

Choose a value on the interval $0 < x < 1.64153795$ and see if this value makes the original inequality true.

$x = 1$

Step 10.4.2.2

Replace $x$ with $1$ in the original inequality.

$(1) (- 16 {(1)}^{3} + 9 (1) + 56) \geq 0$

Step 10.4.2.3

The left side $49$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 10.4.3

Test a value on the interval $x > 1.64153795$ to see if it makes the inequality true.

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Step 10.4.3.1

Choose a value on the interval $x > 1.64153795$ and see if this value makes the original inequality true.

$x = 4$

Step 10.4.3.2

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

$(4) (- 16 {(4)}^{3} + 9 (4) + 56) \geq 0$

Step 10.4.3.3

The left side $- 3728$ is less than the right side $0$ , which means that the given statement is false.

False

Step 10.4.4

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

$x < 0$ False

$0 < x < 1.64153795$ True

$x > 1.64153795$ False

$x < 0$ False

$0 < x < 1.64153795$ True

$x > 1.64153795$ False

Step 10.5

The solution consists of all of the true intervals.

$0 \leq x \leq 1.64153795$

Step 11

Set the denominator in $\frac{3 x - \sqrt{x (- 16 x^{3} + 9 x + 56)}}{4 x}$ equal to $0$ to find where the expression is undefined.

$4 x = 0$

Step 12

Divide each term in $4 x = 0$ by $4$ and simplify.

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Step 12.1

Divide each term in $4 x = 0$ by $4$ .

$\frac{4 x}{4} = \frac{0}{4}$

Step 12.2

Simplify the left side.

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Step 12.2.1

Cancel the common factor of $4$ .

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Step 12.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{0}{4}$

Step 12.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{4}$

Step 12.3

Simplify the right side.

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Step 12.3.1

Divide $0$ by $4$ .

$x = 0$

Step 13

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

Interval Notation:

$(0, 1.64153795]$

Set-Builder Notation:

${x | 0 < x \leq 1.64153795}$

Step 14