Finite Math Examples

$10 x - 25 - x^{2} < 0$

Step 1

Convert the inequality to an equation.

$10 x - 25 - x^{2} = 0$

Step 2

Factor the left side of the equation.

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Step 2.1

Factor $- 1$ out of $10 x - 25 - x^{2}$ .

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Step 2.1.1

Reorder the expression.

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Step 2.1.1.1

Move $- 25$ .

$10 x - x^{2} - 25 = 0$

Step 2.1.1.2

Reorder $10 x$ and $- x^{2}$ .

$- x^{2} + 10 x - 25 = 0$

Step 2.1.2

Factor $- 1$ out of $- x^{2}$ .

$- (x^{2}) + 10 x - 25 = 0$

Step 2.1.3

Factor $- 1$ out of $10 x$ .

$- (x^{2}) - (- 10 x) - 25 = 0$

Step 2.1.4

Rewrite $- 25$ as $- 1 (25)$ .

$- (x^{2}) - (- 10 x) - 1 \cdot 25 = 0$

Step 2.1.5

Factor $- 1$ out of $- (x^{2}) - (- 10 x)$ .

$- (x^{2} - 10 x) - 1 \cdot 25 = 0$

Step 2.1.6

Factor $- 1$ out of $- (x^{2} - 10 x) - 1 (25)$ .

$- (x^{2} - 10 x + 25) = 0$

Step 2.2

Factor using the perfect square rule.

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Step 2.2.1

Rewrite $25$ as $5^{2}$ .

$- (x^{2} - 10 x + 5^{2}) = 0$

Step 2.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$10 x = 2 \cdot x \cdot 5$

Step 2.2.3

Rewrite the polynomial.

$- (x^{2} - 2 \cdot x \cdot 5 + 5^{2}) = 0$

Step 2.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 5$ .

$- {(x - 5)}^{2} = 0$

Step 3

Divide each term in $- {(x - 5)}^{2} = 0$ by $- 1$ and simplify.

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Step 3.1

Divide each term in $- {(x - 5)}^{2} = 0$ by $- 1$ .

$\frac{- {(x - 5)}^{2}}{- 1} = \frac{0}{- 1}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Dividing two negative values results in a positive value.

$\frac{{(x - 5)}^{2}}{1} = \frac{0}{- 1}$

Step 3.2.2

Divide ${(x - 5)}^{2}$ by $1$ .

${(x - 5)}^{2} = \frac{0}{- 1}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $0$ by $- 1$ .

${(x - 5)}^{2} = 0$

Step 4

Set the $x - 5$ equal to $0$ .

$x - 5 = 0$

Step 5

Add $5$ to both sides of the equation.

$x = 5$

Step 6

Use each root to create test intervals.

$x < 5$

$x > 5$

Step 7

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 7.1

Test a value on the interval $x < 5$ to see if it makes the inequality true.

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Step 7.1.1

Choose a value on the interval $x < 5$ and see if this value makes the original inequality true.

$x = 0$

Step 7.1.2

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

$10 (0) - 25 - {(0)}^{2} < 0$

Step 7.1.3

The left side $- 25$ is less than the right side $0$ , which means that the given statement is always true.

True

Step 7.2

Test a value on the interval $x > 5$ to see if it makes the inequality true.

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Step 7.2.1

Choose a value on the interval $x > 5$ and see if this value makes the original inequality true.

$x = 8$

Step 7.2.2

Replace $x$ with $8$ in the original inequality.

$10 (8) - 25 - {(8)}^{2} < 0$

Step 7.2.3

The left side $- 9$ is less than the right side $0$ , which means that the given statement is always true.

True

Step 7.3

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

$x < 5$ True

$x > 5$ True

$x < 5$ True

$x > 5$ True

Step 8

The solution consists of all of the true intervals.

$x < 5$ or $x > 5$

Step 9

Convert the inequality to interval notation.

$(- \infty, 5) \cup (5, \infty)$

Step 10