Finite Math Examples

20000 < - 2 x^{2} + 640 x < 40000

$20000 < - 2 x^{2} + 640 x < 40000$

Step 1

Divide each term in the inequality by

$- 2$ .

$\frac{20000}{- 2} > \frac{- 2 x^{2}}{- 2} + \frac{640 x}{- 2} > \frac{40000}{- 2}$

Step 2

Divide

$20000$ by

$- 2$ .

$- 10000 > \frac{- 2 x^{2}}{- 2} + \frac{640 x}{- 2} > \frac{40000}{- 2}$

Step 3

Simplify each term.

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Step 3.1

Cancel the common factor of

$- 2$ .

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Step 3.1.1

Cancel the common factor.

$- 10000 > \frac{- 2 x^{2}}{- 2} + \frac{640 x}{- 2} > \frac{40000}{- 2}$

Step 3.1.2

Divide

$x^{2}$ by

$1$ .

$- 10000 > x^{2} + \frac{640 x}{- 2} > \frac{40000}{- 2}$

Step 3.2

Cancel the common factor of

$640$ and

$- 2$ .

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Step 3.2.1

Factor

$2$ out of

$640 x$ .

$- 10000 > x^{2} + \frac{2 (320 x)}{- 2} > \frac{40000}{- 2}$

Step 3.2.2

Move the negative one from the denominator of

$\frac{320 x}{- 1}$ .

$- 10000 > x^{2} - 1 \cdot (320 x) > \frac{40000}{- 2}$

Step 3.3

Rewrite

$- 1 \cdot (320 x)$ as

$- (320 x)$ .

$- 10000 > x^{2} - (320 x) > \frac{40000}{- 2}$

Step 3.4

Multiply

$320$ by

$- 1$ .

$- 10000 > x^{2} - 320 x > \frac{40000}{- 2}$

Step 4

Divide

$40000$ by

$- 2$ .

$- 10000 > x^{2} - 320 x > - 20000$

Step 5

To isolate a single

$x$ variable, take the root of degree

$2$ of each expression.

$\sqrt{- 10000} > \sqrt{x^{2} - 320 x} > \sqrt{- 20000}$

Step 6

Rewrite

$- 10000$ as

$- 1 (10000)$ .

$\sqrt{- 1 \cdot 10000} > \sqrt{x^{2} - 320 x} > \sqrt{- 20000}$

Step 7

Rewrite

$\sqrt{- 1 (10000)}$ as

$\sqrt{- 1} \cdot \sqrt{10000}$ .

$\sqrt{- 1} \cdot \sqrt{10000} > \sqrt{x^{2} - 320 x} > \sqrt{- 20000}$

Step 8

Rewrite

$\sqrt{- 1}$ as

$i$ .

$i \cdot \sqrt{10000} > \sqrt{x^{2} - 320 x} > \sqrt{- 20000}$

Step 9

Rewrite

$10000$ as

$100^{2}$ .

$i \cdot \sqrt{100^{2}} > \sqrt{x^{2} - 320 x} > \sqrt{- 20000}$

Step 10

Pull terms out from under the radical.

$i \cdot | 100 | > \sqrt{x^{2} - 320 x} > \sqrt{- 20000}$

Step 11

The absolute value is the distance between a number and zero. The distance between

$0$ and

$100$ is

$100$ .

$i \cdot 100 > \sqrt{x^{2} - 320 x} > \sqrt{- 20000}$

Step 12

Move

$100$ to the left of

$i$ .

$100 i > \sqrt{x^{2} - 320 x} > \sqrt{- 20000}$

Step 13

Factor

$x$ out of

$x^{2} - 320 x$ .

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Step 13.1

Factor

$x$ out of

$x^{2}$ .

$100 i > \sqrt{x \cdot x - 320 x} > \sqrt{- 20000}$

Step 13.2

Factor

$x$ out of

$- 320 x$ .

$100 i > \sqrt{x \cdot x + x \cdot - 320} > \sqrt{- 20000}$

Step 13.3

Factor

$x$ out of

$x \cdot x + x \cdot - 320$ .

$100 i > \sqrt{x (x - 320)} > \sqrt{- 20000}$

Step 14

Rewrite

$- 20000$ as

$- 1 (20000)$ .

$100 i > \sqrt{x (x - 320)} > \sqrt{- 1 \cdot 20000}$

Step 15

Rewrite

$\sqrt{- 1 (20000)}$ as

$\sqrt{- 1} \cdot \sqrt{20000}$ .

$100 i > \sqrt{x (x - 320)} > \sqrt{- 1} \cdot \sqrt{20000}$

Step 16

Rewrite

$\sqrt{- 1}$ as

$i$ .

$100 i > \sqrt{x (x - 320)} > i \cdot \sqrt{20000}$

Step 17

Rewrite

$20000$ as

$100^{2} \cdot 2$ .

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Step 17.1

Factor

$10000$ out of

$20000$ .

$100 i > \sqrt{x (x - 320)} > i \cdot \sqrt{10000 (2)}$

Step 17.2

Rewrite

$10000$ as

$100^{2}$ .

$100 i > \sqrt{x (x - 320)} > i \cdot \sqrt{100^{2} \cdot 2}$

Step 18

Pull terms out from under the radical.

$100 i > \sqrt{x (x - 320)} > i \cdot (| 100 | \sqrt{2})$

Step 19

The absolute value is the distance between a number and zero. The distance between

$0$ and

$100$ is

$100$ .

$100 i > \sqrt{x (x - 320)} > i \cdot (100 \sqrt{2})$

Step 20

Move

$100$ to the left of

$i$ .

$100 i > \sqrt{x (x - 320)} > 100 i \sqrt{2}$

Step 21

Convert the inequality to interval notation.

No solution