Finite Math Examples

${(x + 7)}^{2} > 50 x - 27$

Step 1

Move all terms containing $x$ to the left side of the inequality.

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Step 1.1

Subtract $50 x$ from both sides of the inequality.

${(x + 7)}^{2} - 50 x > - 27$

Step 1.2

Simplify each term.

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Step 1.2.1

Rewrite ${(x + 7)}^{2}$ as $(x + 7) (x + 7)$ .

$(x + 7) (x + 7) - 50 x > - 27$

Step 1.2.2

Expand $(x + 7) (x + 7)$ using the FOIL Method.

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Step 1.2.2.1

Apply the distributive property.

$x (x + 7) + 7 (x + 7) - 50 x > - 27$

Step 1.2.2.2

Apply the distributive property.

$x \cdot x + x \cdot 7 + 7 (x + 7) - 50 x > - 27$

Step 1.2.2.3

Apply the distributive property.

$x \cdot x + x \cdot 7 + 7 x + 7 \cdot 7 - 50 x > - 27$

Step 1.2.3

Simplify and combine like terms.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot 7 + 7 x + 7 \cdot 7 - 50 x > - 27$

Step 1.2.3.1.2

Move $7$ to the left of $x$ .

$x^{2} + 7 \cdot x + 7 x + 7 \cdot 7 - 50 x > - 27$

Step 1.2.3.1.3

Multiply $7$ by $7$ .

$x^{2} + 7 x + 7 x + 49 - 50 x > - 27$

Step 1.2.3.2

Add $7 x$ and $7 x$ .

$x^{2} + 14 x + 49 - 50 x > - 27$

Step 1.3

Subtract $50 x$ from $14 x$ .

$x^{2} - 36 x + 49 > - 27$

Step 2

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

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Step 2.1

Add $27$ to both sides of the inequality.

$x^{2} - 36 x + 49 + 27 > 0$

Step 2.2

Add $49$ and $27$ .

$x^{2} - 36 x + 76 > 0$

Step 3

Convert the inequality to an equation.

$x^{2} - 36 x + 76 = 0$

Step 4

Use the quadratic formula to find the solutions.

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

Step 5

Substitute the values $a = 1$ , $b = - 36$ , and $c = 76$ into the quadratic formula and solve for $x$ .

$\frac{36 \pm \sqrt{{(- 36)}^{2} - 4 \cdot (1 \cdot 76)}}{2 \cdot 1}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $- 36$ to the power of $2$ .

$x = \frac{36 \pm \sqrt{1296 - 4 \cdot 1 \cdot 76}}{2 \cdot 1}$

Step 6.1.2

Multiply $- 4 \cdot 1 \cdot 76$ .

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Step 6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{36 \pm \sqrt{1296 - 4 \cdot 76}}{2 \cdot 1}$

Step 6.1.2.2

Multiply $- 4$ by $76$ .

$x = \frac{36 \pm \sqrt{1296 - 304}}{2 \cdot 1}$

Step 6.1.3

Subtract $304$ from $1296$ .

$x = \frac{36 \pm \sqrt{992}}{2 \cdot 1}$

Step 6.1.4

Rewrite $992$ as $4^{2} \cdot 62$ .

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Step 6.1.4.1

Factor $16$ out of $992$ .

$x = \frac{36 \pm \sqrt{16 (62)}}{2 \cdot 1}$

Step 6.1.4.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{36 \pm \sqrt{4^{2} \cdot 62}}{2 \cdot 1}$

Step 6.1.5

Pull terms out from under the radical.

$x = \frac{36 \pm 4 \sqrt{62}}{2 \cdot 1}$

Step 6.2

Multiply $2$ by $1$ .

$x = \frac{36 \pm 4 \sqrt{62}}{2}$

Step 6.3

Simplify $\frac{36 \pm 4 \sqrt{62}}{2}$ .

$x = 18 \pm 2 \sqrt{62}$

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

Raise $- 36$ to the power of $2$ .

$x = \frac{36 \pm \sqrt{1296 - 4 \cdot 1 \cdot 76}}{2 \cdot 1}$

Step 7.1.2

Multiply $- 4 \cdot 1 \cdot 76$ .

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Step 7.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{36 \pm \sqrt{1296 - 4 \cdot 76}}{2 \cdot 1}$

Step 7.1.2.2

Multiply $- 4$ by $76$ .

$x = \frac{36 \pm \sqrt{1296 - 304}}{2 \cdot 1}$

Step 7.1.3

Subtract $304$ from $1296$ .

$x = \frac{36 \pm \sqrt{992}}{2 \cdot 1}$

Step 7.1.4

Rewrite $992$ as $4^{2} \cdot 62$ .

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Step 7.1.4.1

Factor $16$ out of $992$ .

$x = \frac{36 \pm \sqrt{16 (62)}}{2 \cdot 1}$

Step 7.1.4.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{36 \pm \sqrt{4^{2} \cdot 62}}{2 \cdot 1}$

Step 7.1.5

Pull terms out from under the radical.

$x = \frac{36 \pm 4 \sqrt{62}}{2 \cdot 1}$

Step 7.2

Multiply $2$ by $1$ .

$x = \frac{36 \pm 4 \sqrt{62}}{2}$

Step 7.3

Simplify $\frac{36 \pm 4 \sqrt{62}}{2}$ .

$x = 18 \pm 2 \sqrt{62}$

Step 7.4

Change the $\pm$ to $+$ .

$x = 18 + 2 \sqrt{62}$

Step 8

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

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise $- 36$ to the power of $2$ .

$x = \frac{36 \pm \sqrt{1296 - 4 \cdot 1 \cdot 76}}{2 \cdot 1}$

Step 8.1.2

Multiply $- 4 \cdot 1 \cdot 76$ .

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Step 8.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{36 \pm \sqrt{1296 - 4 \cdot 76}}{2 \cdot 1}$

Step 8.1.2.2

Multiply $- 4$ by $76$ .

$x = \frac{36 \pm \sqrt{1296 - 304}}{2 \cdot 1}$

Step 8.1.3

Subtract $304$ from $1296$ .

$x = \frac{36 \pm \sqrt{992}}{2 \cdot 1}$

Step 8.1.4

Rewrite $992$ as $4^{2} \cdot 62$ .

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Step 8.1.4.1

Factor $16$ out of $992$ .

$x = \frac{36 \pm \sqrt{16 (62)}}{2 \cdot 1}$

Step 8.1.4.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{36 \pm \sqrt{4^{2} \cdot 62}}{2 \cdot 1}$

Step 8.1.5

Pull terms out from under the radical.

$x = \frac{36 \pm 4 \sqrt{62}}{2 \cdot 1}$

Step 8.2

Multiply $2$ by $1$ .

$x = \frac{36 \pm 4 \sqrt{62}}{2}$

Step 8.3

Simplify $\frac{36 \pm 4 \sqrt{62}}{2}$ .

$x = 18 \pm 2 \sqrt{62}$

Step 8.4

Change the $\pm$ to $-$ .

$x = 18 - 2 \sqrt{62}$

Step 9

Consolidate the solutions.

$x = 18 + 2 \sqrt{62}, 18 - 2 \sqrt{62}$

Step 10

Use each root to create test intervals.

$x < 18 - 2 \sqrt{62}$

$18 - 2 \sqrt{62} < x < 18 + 2 \sqrt{62}$

$x > 18 + 2 \sqrt{62}$

Step 11

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 11.1

Test a value on the interval $x < 18 - 2 \sqrt{62}$ to see if it makes the inequality true.

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Step 11.1.1

Choose a value on the interval $x < 18 - 2 \sqrt{62}$ and see if this value makes the original inequality true.

$x = 0$

Step 11.1.2

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

${((0) + 7)}^{2} > 50 (0) - 27$

Step 11.1.3

The left side $49$ is greater than the right side $- 27$ , which means that the given statement is always true.

True

Step 11.2

Test a value on the interval $18 - 2 \sqrt{62} < x < 18 + 2 \sqrt{62}$ to see if it makes the inequality true.

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Step 11.2.1

Choose a value on the interval $18 - 2 \sqrt{62} < x < 18 + 2 \sqrt{62}$ and see if this value makes the original inequality true.

$x = 5$

Step 11.2.2

Replace $x$ with $5$ in the original inequality.

${((5) + 7)}^{2} > 50 (5) - 27$

Step 11.2.3

The left side $144$ is not greater than the right side $223$ , which means that the given statement is false.

False

Step 11.3

Test a value on the interval $x > 18 + 2 \sqrt{62}$ to see if it makes the inequality true.

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Step 11.3.1

Choose a value on the interval $x > 18 + 2 \sqrt{62}$ and see if this value makes the original inequality true.

$x = 36$

Step 11.3.2

Replace $x$ with $36$ in the original inequality.

${((36) + 7)}^{2} > 50 (36) - 27$

Step 11.3.3

The left side $1849$ is greater than the right side $1773$ , which means that the given statement is always true.

True

Step 11.4

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

$x < 18 - 2 \sqrt{62}$ True

$18 - 2 \sqrt{62} < x < 18 + 2 \sqrt{62}$ False

$x > 18 + 2 \sqrt{62}$ True

$x < 18 - 2 \sqrt{62}$ True

$18 - 2 \sqrt{62} < x < 18 + 2 \sqrt{62}$ False

$x > 18 + 2 \sqrt{62}$ True

Step 12

The solution consists of all of the true intervals.

$x < 18 - 2 \sqrt{62}$ or $x > 18 + 2 \sqrt{62}$

Step 13

Convert the inequality to interval notation.

$(- \infty, 18 - 2 \sqrt{62}) \cup (18 + 2 \sqrt{62}, \infty)$

Step 14