Linear Algebra Examples

$x^{2} + y^{2} - 8 x + 14 y + 65 = 36$

Step 1

Move all terms to the left side of the equation and simplify.

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Step 1.1

Subtract $36$ from both sides of the equation.

$x^{2} + y^{2} - 8 x + 14 y + 65 - 36 = 0$

Step 1.2

Subtract $36$ from $65$ .

$x^{2} + y^{2} - 8 x + 14 y + 29 = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = 1$ , $b = 14$ , and $c = x^{2} - 8 x + 29$ into the quadratic formula and solve for $y$ .

$\frac{- 14 \pm \sqrt{14^{2} - 4 \cdot (1 \cdot (x^{2} - 8 x + 29))}}{2 \cdot 1}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $14$ to the power of $2$ .

$y = \frac{- 14 \pm \sqrt{196 - 4 \cdot 1 \cdot (x^{2} - 8 x + 29)}}{2 \cdot 1}$

Step 4.1.2

Multiply $- 4$ by $1$ .

$y = \frac{- 14 \pm \sqrt{196 - 4 \cdot (x^{2} - 8 x + 29)}}{2 \cdot 1}$

Step 4.1.3

Apply the distributive property.

$y = \frac{- 14 \pm \sqrt{196 - 4 x^{2} - 4 (- 8 x) - 4 \cdot 29}}{2 \cdot 1}$

Step 4.1.4

Simplify.

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Step 4.1.4.1

Multiply $- 8$ by $- 4$ .

$y = \frac{- 14 \pm \sqrt{196 - 4 x^{2} + 32 x - 4 \cdot 29}}{2 \cdot 1}$

Step 4.1.4.2

Multiply $- 4$ by $29$ .

$y = \frac{- 14 \pm \sqrt{196 - 4 x^{2} + 32 x - 116}}{2 \cdot 1}$

Step 4.1.5

Subtract $116$ from $196$ .

$y = \frac{- 14 \pm \sqrt{- 4 x^{2} + 32 x + 80}}{2 \cdot 1}$

Step 4.1.6

Rewrite $- 4 x^{2} + 32 x + 80$ in a factored form.

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Step 4.1.6.1

Factor $4$ out of $- 4 x^{2} + 32 x + 80$ .

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Step 4.1.6.1.1

Factor $4$ out of $- 4 x^{2}$ .

$y = \frac{- 14 \pm \sqrt{4 (- x^{2}) + 32 x + 80}}{2 \cdot 1}$

Step 4.1.6.1.2

Factor $4$ out of $32 x$ .

$y = \frac{- 14 \pm \sqrt{4 (- x^{2}) + 4 (8 x) + 80}}{2 \cdot 1}$

Step 4.1.6.1.3

Factor $4$ out of $80$ .

$y = \frac{- 14 \pm \sqrt{4 (- x^{2}) + 4 (8 x) + 4 (20)}}{2 \cdot 1}$

Step 4.1.6.1.4

Factor $4$ out of $4 (- x^{2}) + 4 (8 x)$ .

$y = \frac{- 14 \pm \sqrt{4 (- x^{2} + 8 x) + 4 (20)}}{2 \cdot 1}$

Step 4.1.6.1.5

Factor $4$ out of $4 (- x^{2} + 8 x) + 4 (20)$ .

$y = \frac{- 14 \pm \sqrt{4 (- x^{2} + 8 x + 20)}}{2 \cdot 1}$

Step 4.1.6.2

Factor by grouping.

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Step 4.1.6.2.1

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 $a \cdot c = - 1 \cdot 20 = - 20$ and whose sum is $b = 8$ .

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Step 4.1.6.2.1.1

Factor $8$ out of $8 x$ .

$y = \frac{- 14 \pm \sqrt{4 (- x^{2} + 8 (x) + 20)}}{2 \cdot 1}$

Step 4.1.6.2.1.2

Rewrite $8$ as $- 2$ plus $10$

$y = \frac{- 14 \pm \sqrt{4 (- x^{2} + (- 2 + 10) x + 20)}}{2 \cdot 1}$

Step 4.1.6.2.1.3

Apply the distributive property.

$y = \frac{- 14 \pm \sqrt{4 (- x^{2} - 2 x + 10 x + 20)}}{2 \cdot 1}$

Step 4.1.6.2.2

Factor out the greatest common factor from each group.

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Step 4.1.6.2.2.1

Group the first two terms and the last two terms.

$y = \frac{- 14 \pm \sqrt{4 ((- x^{2} - 2 x) + 10 x + 20)}}{2 \cdot 1}$

Step 4.1.6.2.2.2

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

$y = \frac{- 14 \pm \sqrt{4 (x (- x - 2) - 10 (- x - 2))}}{2 \cdot 1}$

Step 4.1.6.2.3

Factor the polynomial by factoring out the greatest common factor, $- x - 2$ .

$y = \frac{- 14 \pm \sqrt{4 ((- x - 2) (x - 10))}}{2 \cdot 1}$

$y = \frac{- 14 \pm \sqrt{4 (- x - 2) (x - 10)}}{2 \cdot 1}$

Step 4.1.7

Rewrite $4 (- x - 2) (x - 10)$ as $2^{2} ((- x - 2) (x - 10))$ .

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Step 4.1.7.1

Rewrite $4$ as $2^{2}$ .

$y = \frac{- 14 \pm \sqrt{2^{2} (- x - 2) (x - 10)}}{2 \cdot 1}$

Step 4.1.7.2

Add parentheses.

$y = \frac{- 14 \pm \sqrt{2^{2} ((- x - 2) (x - 10))}}{2 \cdot 1}$

Step 4.1.8

Pull terms out from under the radical.

$y = \frac{- 14 \pm 2 \sqrt{(- x - 2) (x - 10)}}{2 \cdot 1}$

Step 4.2

Multiply $2$ by $1$ .

$y = \frac{- 14 \pm 2 \sqrt{(- x - 2) (x - 10)}}{2}$

Step 4.3

Simplify $\frac{- 14 \pm 2 \sqrt{(- x - 2) (x - 10)}}{2}$ .

$y = - 7 \pm \sqrt{(- x - 2) (x - 10)}$

Step 5

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

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $14$ to the power of $2$ .

$y = \frac{- 14 \pm \sqrt{196 - 4 \cdot 1 \cdot (x^{2} - 8 x + 29)}}{2 \cdot 1}$

Step 5.1.2

Multiply $- 4$ by $1$ .

$y = \frac{- 14 \pm \sqrt{196 - 4 \cdot (x^{2} - 8 x + 29)}}{2 \cdot 1}$

Step 5.1.3

Apply the distributive property.

$y = \frac{- 14 \pm \sqrt{196 - 4 x^{2} - 4 (- 8 x) - 4 \cdot 29}}{2 \cdot 1}$

Step 5.1.4

Simplify.

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Step 5.1.4.1

Multiply $- 8$ by $- 4$ .

$y = \frac{- 14 \pm \sqrt{196 - 4 x^{2} + 32 x - 4 \cdot 29}}{2 \cdot 1}$

Step 5.1.4.2

Multiply $- 4$ by $29$ .

$y = \frac{- 14 \pm \sqrt{196 - 4 x^{2} + 32 x - 116}}{2 \cdot 1}$

Step 5.1.5

Subtract $116$ from $196$ .

$y = \frac{- 14 \pm \sqrt{- 4 x^{2} + 32 x + 80}}{2 \cdot 1}$

Step 5.1.6

Rewrite $- 4 x^{2} + 32 x + 80$ in a factored form.

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Step 5.1.6.1

Factor $4$ out of $- 4 x^{2} + 32 x + 80$ .

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Step 5.1.6.1.1

Factor $4$ out of $- 4 x^{2}$ .

$y = \frac{- 14 \pm \sqrt{4 (- x^{2}) + 32 x + 80}}{2 \cdot 1}$

Step 5.1.6.1.2

Factor $4$ out of $32 x$ .

$y = \frac{- 14 \pm \sqrt{4 (- x^{2}) + 4 (8 x) + 80}}{2 \cdot 1}$

Step 5.1.6.1.3

Factor $4$ out of $80$ .

$y = \frac{- 14 \pm \sqrt{4 (- x^{2}) + 4 (8 x) + 4 (20)}}{2 \cdot 1}$

Step 5.1.6.1.4

Factor $4$ out of $4 (- x^{2}) + 4 (8 x)$ .

$y = \frac{- 14 \pm \sqrt{4 (- x^{2} + 8 x) + 4 (20)}}{2 \cdot 1}$

Step 5.1.6.1.5

Factor $4$ out of $4 (- x^{2} + 8 x) + 4 (20)$ .

$y = \frac{- 14 \pm \sqrt{4 (- x^{2} + 8 x + 20)}}{2 \cdot 1}$

Step 5.1.6.2

Factor by grouping.

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Step 5.1.6.2.1

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 $a \cdot c = - 1 \cdot 20 = - 20$ and whose sum is $b = 8$ .

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Step 5.1.6.2.1.1

Factor $8$ out of $8 x$ .

$y = \frac{- 14 \pm \sqrt{4 (- x^{2} + 8 (x) + 20)}}{2 \cdot 1}$

Step 5.1.6.2.1.2

Rewrite $8$ as $- 2$ plus $10$

$y = \frac{- 14 \pm \sqrt{4 (- x^{2} + (- 2 + 10) x + 20)}}{2 \cdot 1}$

Step 5.1.6.2.1.3

Apply the distributive property.

$y = \frac{- 14 \pm \sqrt{4 (- x^{2} - 2 x + 10 x + 20)}}{2 \cdot 1}$

Step 5.1.6.2.2

Factor out the greatest common factor from each group.

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Step 5.1.6.2.2.1

Group the first two terms and the last two terms.

$y = \frac{- 14 \pm \sqrt{4 ((- x^{2} - 2 x) + 10 x + 20)}}{2 \cdot 1}$

Step 5.1.6.2.2.2

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

$y = \frac{- 14 \pm \sqrt{4 (x (- x - 2) - 10 (- x - 2))}}{2 \cdot 1}$

Step 5.1.6.2.3

Factor the polynomial by factoring out the greatest common factor, $- x - 2$ .

$y = \frac{- 14 \pm \sqrt{4 ((- x - 2) (x - 10))}}{2 \cdot 1}$

$y = \frac{- 14 \pm \sqrt{4 (- x - 2) (x - 10)}}{2 \cdot 1}$

Step 5.1.7

Rewrite $4 (- x - 2) (x - 10)$ as $2^{2} ((- x - 2) (x - 10))$ .

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Step 5.1.7.1

Rewrite $4$ as $2^{2}$ .

$y = \frac{- 14 \pm \sqrt{2^{2} (- x - 2) (x - 10)}}{2 \cdot 1}$

Step 5.1.7.2

Add parentheses.

$y = \frac{- 14 \pm \sqrt{2^{2} ((- x - 2) (x - 10))}}{2 \cdot 1}$

Step 5.1.8

Pull terms out from under the radical.

$y = \frac{- 14 \pm 2 \sqrt{(- x - 2) (x - 10)}}{2 \cdot 1}$

Step 5.2

Multiply $2$ by $1$ .

$y = \frac{- 14 \pm 2 \sqrt{(- x - 2) (x - 10)}}{2}$

Step 5.3

Simplify $\frac{- 14 \pm 2 \sqrt{(- x - 2) (x - 10)}}{2}$ .

$y = - 7 \pm \sqrt{(- x - 2) (x - 10)}$

Step 5.4

Change the $\pm$ to $+$ .

$y = - 7 + \sqrt{(- x - 2) (x - 10)}$

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

Raise $14$ to the power of $2$ .

$y = \frac{- 14 \pm \sqrt{196 - 4 \cdot 1 \cdot (x^{2} - 8 x + 29)}}{2 \cdot 1}$

Step 6.1.2

Multiply $- 4$ by $1$ .

$y = \frac{- 14 \pm \sqrt{196 - 4 \cdot (x^{2} - 8 x + 29)}}{2 \cdot 1}$

Step 6.1.3

Apply the distributive property.

$y = \frac{- 14 \pm \sqrt{196 - 4 x^{2} - 4 (- 8 x) - 4 \cdot 29}}{2 \cdot 1}$

Step 6.1.4

Simplify.

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Step 6.1.4.1

Multiply $- 8$ by $- 4$ .

$y = \frac{- 14 \pm \sqrt{196 - 4 x^{2} + 32 x - 4 \cdot 29}}{2 \cdot 1}$

Step 6.1.4.2

Multiply $- 4$ by $29$ .

$y = \frac{- 14 \pm \sqrt{196 - 4 x^{2} + 32 x - 116}}{2 \cdot 1}$

Step 6.1.5

Subtract $116$ from $196$ .

$y = \frac{- 14 \pm \sqrt{- 4 x^{2} + 32 x + 80}}{2 \cdot 1}$

Step 6.1.6

Rewrite $- 4 x^{2} + 32 x + 80$ in a factored form.

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Step 6.1.6.1

Factor $4$ out of $- 4 x^{2} + 32 x + 80$ .

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Step 6.1.6.1.1

Factor $4$ out of $- 4 x^{2}$ .

$y = \frac{- 14 \pm \sqrt{4 (- x^{2}) + 32 x + 80}}{2 \cdot 1}$

Step 6.1.6.1.2

Factor $4$ out of $32 x$ .

$y = \frac{- 14 \pm \sqrt{4 (- x^{2}) + 4 (8 x) + 80}}{2 \cdot 1}$

Step 6.1.6.1.3

Factor $4$ out of $80$ .

$y = \frac{- 14 \pm \sqrt{4 (- x^{2}) + 4 (8 x) + 4 (20)}}{2 \cdot 1}$

Step 6.1.6.1.4

Factor $4$ out of $4 (- x^{2}) + 4 (8 x)$ .

$y = \frac{- 14 \pm \sqrt{4 (- x^{2} + 8 x) + 4 (20)}}{2 \cdot 1}$

Step 6.1.6.1.5

Factor $4$ out of $4 (- x^{2} + 8 x) + 4 (20)$ .

$y = \frac{- 14 \pm \sqrt{4 (- x^{2} + 8 x + 20)}}{2 \cdot 1}$

Step 6.1.6.2

Factor by grouping.

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Step 6.1.6.2.1

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 $a \cdot c = - 1 \cdot 20 = - 20$ and whose sum is $b = 8$ .

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Step 6.1.6.2.1.1

Factor $8$ out of $8 x$ .

$y = \frac{- 14 \pm \sqrt{4 (- x^{2} + 8 (x) + 20)}}{2 \cdot 1}$

Step 6.1.6.2.1.2

Rewrite $8$ as $- 2$ plus $10$

$y = \frac{- 14 \pm \sqrt{4 (- x^{2} + (- 2 + 10) x + 20)}}{2 \cdot 1}$

Step 6.1.6.2.1.3

Apply the distributive property.

$y = \frac{- 14 \pm \sqrt{4 (- x^{2} - 2 x + 10 x + 20)}}{2 \cdot 1}$

Step 6.1.6.2.2

Factor out the greatest common factor from each group.

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Step 6.1.6.2.2.1

Group the first two terms and the last two terms.

$y = \frac{- 14 \pm \sqrt{4 ((- x^{2} - 2 x) + 10 x + 20)}}{2 \cdot 1}$

Step 6.1.6.2.2.2

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

$y = \frac{- 14 \pm \sqrt{4 (x (- x - 2) - 10 (- x - 2))}}{2 \cdot 1}$

Step 6.1.6.2.3

Factor the polynomial by factoring out the greatest common factor, $- x - 2$ .

$y = \frac{- 14 \pm \sqrt{4 ((- x - 2) (x - 10))}}{2 \cdot 1}$

$y = \frac{- 14 \pm \sqrt{4 (- x - 2) (x - 10)}}{2 \cdot 1}$

Step 6.1.7

Rewrite $4 (- x - 2) (x - 10)$ as $2^{2} ((- x - 2) (x - 10))$ .

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Step 6.1.7.1

Rewrite $4$ as $2^{2}$ .

$y = \frac{- 14 \pm \sqrt{2^{2} (- x - 2) (x - 10)}}{2 \cdot 1}$

Step 6.1.7.2

Add parentheses.

$y = \frac{- 14 \pm \sqrt{2^{2} ((- x - 2) (x - 10))}}{2 \cdot 1}$

Step 6.1.8

Pull terms out from under the radical.

$y = \frac{- 14 \pm 2 \sqrt{(- x - 2) (x - 10)}}{2 \cdot 1}$

Step 6.2

Multiply $2$ by $1$ .

$y = \frac{- 14 \pm 2 \sqrt{(- x - 2) (x - 10)}}{2}$

Step 6.3

Simplify $\frac{- 14 \pm 2 \sqrt{(- x - 2) (x - 10)}}{2}$ .

$y = - 7 \pm \sqrt{(- x - 2) (x - 10)}$

Step 6.4

Change the $\pm$ to $-$ .

$y = - 7 - \sqrt{(- x - 2) (x - 10)}$

Step 7

The final answer is the combination of both solutions.

$y = - 7 + \sqrt{(- x - 2) (x - 10)}$

$y = - 7 - \sqrt{(- x - 2) (x - 10)}$

Step 8

Set the radicand in $\sqrt{(- x - 2) (x - 10)}$ greater than or equal to $0$ to find where the expression is defined.

$(- x - 2) (x - 10) \geq 0$

Step 9

Solve for $x$ .

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Step 9.1

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

$- x - 2 = 0$

$x - 10 = 0$

Step 9.2

Set $- x - 2$ equal to $0$ and solve for $x$ .

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Step 9.2.1

Set $- x - 2$ equal to $0$ .

$- x - 2 = 0$

Step 9.2.2

Solve $- x - 2 = 0$ for $x$ .

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Step 9.2.2.1

Add $2$ to both sides of the equation.

$- x = 2$

Step 9.2.2.2

Divide each term in $- x = 2$ by $- 1$ and simplify.

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Step 9.2.2.2.1

Divide each term in $- x = 2$ by $- 1$ .

$\frac{- x}{- 1} = \frac{2}{- 1}$

Step 9.2.2.2.2

Simplify the left side.

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Step 9.2.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{2}{- 1}$

Step 9.2.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{2}{- 1}$

Step 9.2.2.2.3

Simplify the right side.

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Step 9.2.2.2.3.1

Divide $2$ by $- 1$ .

$x = - 2$

Step 9.3

Set $x - 10$ equal to $0$ and solve for $x$ .

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Step 9.3.1

Set $x - 10$ equal to $0$ .

$x - 10 = 0$

Step 9.3.2

Add $10$ to both sides of the equation.

$x = 10$

Step 9.4

The final solution is all the values that make $(- x - 2) (x - 10) \geq 0$ true.

$x = - 2, 10$

Step 9.5

Use each root to create test intervals.

$x < - 2$

$- 2 < x < 10$

$x > 10$

Step 9.6

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 9.6.1

Test a value on the interval $x < - 2$ to see if it makes the inequality true.

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Step 9.6.1.1

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

$x = - 4$

Step 9.6.1.2

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

$(- (- 4) - 2) ((- 4) - 10) \geq 0$

Step 9.6.1.3

The left side $- 28$ is less than the right side $0$ , which means that the given statement is false.

False

Step 9.6.2

Test a value on the interval $- 2 < x < 10$ to see if it makes the inequality true.

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Step 9.6.2.1

Choose a value on the interval $- 2 < x < 10$ and see if this value makes the original inequality true.

$x = 0$

Step 9.6.2.2

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

$(- (0) - 2) ((0) - 10) \geq 0$

Step 9.6.2.3

The left side $20$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 9.6.3

Test a value on the interval $x > 10$ to see if it makes the inequality true.

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Step 9.6.3.1

Choose a value on the interval $x > 10$ and see if this value makes the original inequality true.

$x = 12$

Step 9.6.3.2

Replace $x$ with $12$ in the original inequality.

$(- (12) - 2) ((12) - 10) \geq 0$

Step 9.6.3.3

The left side $- 28$ is less than the right side $0$ , which means that the given statement is false.

False

Step 9.6.4

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

$x < - 2$ False

$- 2 < x < 10$ True

$x > 10$ False

$x < - 2$ False

$- 2 < x < 10$ True

$x > 10$ False

Step 9.7

The solution consists of all of the true intervals.

$- 2 \leq x \leq 10$

Step 10

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

Interval Notation:

$[- 2, 10]$

Set-Builder Notation:

${x | - 2 \leq x \leq 10}$

Step 11