What is x^2e^x-2e^x<0 ?

The solution to x^2e^x-2e^x<0 is -sqrt(2)<x<sqrt(2)

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x^2e^x-2e^x<0

x^{2} e^{x} - 2 e^{x} < 0

- 2 < x < 2

Interval Notation

(- 2, 2)

Decimal Notation

- 1.41421 \dots < x < 1.41421 \dots

Solution steps

x^{2} e^{x} - 2 e^{x} < 0

Factor out common term

e^{x}

e^{x} (x^{2} - 2)

Factor

x^{2} - 2 : (x + 2) (x - 2)

= e^{x} (x + 2) (x - 2)

Find the signs of the factors of

e^{x} (x + 2) (x - 2)

Identify the intervals:

x < - 2, x = - 2, - 2 < x < 2, x = 2, x > 2

Summarize in a table:

e^{x} x + 2 x - 2 e^{x} (x + 2) (x - 2) x < - 2 + - - + x = - 2 + 0 - 0 - 2 < x < 2 + + - - x = 2 + + 00 x > 2 + + + +

Identify the intervals that satisfy the required condition:

< 0

- 2 < x < 2

\frac{p}{2} - 22 \leq - 21

2 x + 5 < 17

\frac{- 23 + 11 i}{5 + i}

\frac{4}{9} \cdot \frac{9}{4}

49 y^{2} - 49 x^{2}