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

2x² - x - 1 = 0

 

Divide through by 2.

x² - (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 )