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#1**+2 **

First let's find what x values make the expression equal to zero.

2x^{2} - x - 1 = 0

Divide through by 2.

x^{2} - (1/2)x - (1/2) = 0

What two numbers add to -1/2 and multiply to -1/2 ? → -1 and 1/2

So we can factor this

(x - 1)(x + 1/2) = 0

Set each factor equal to zero and solve for x.

x - 1 = 0 x + 1/2 = 0

x = 1 x = -1/2

So, the interval that makes the expression *less* than zero will be

either ( - ∞ , -1/2 ) \(\cup\) ( 1 , ∞ ) or ( -1/2 , 1 )

Test a point in the first interval, say -1 .

2(-1)^{2} - (-1) - 1 < 0

2 < 0 false

Test a point in the second interval, say 0 .

2(0)^{2} - 0 - 1 < 0

-1 < 0 true

So, the x-values which make the expression less than zero must be in the interval ( -1/2 , 1 )

hectictar May 28, 2017