Answer: Anything smaller than -6.25
In other words, c < -6.25
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Step-by-step explanation:
Let's expand out (x - 1)(x + 4)
(x - 1)(x + 4)
w(x + 4) ... let w = x-1
wx + w*4
x( w ) + 4( w )
x(x-1) + 4(x-1) .... replace w with x-1
x^2-x + 4x - 4
x^2+3x-4
Or you could use the FOIL rule if you wanted.
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The graph of y = x^2+3x-4 is a parabola that dips down to some lowest point before going back up again. We're interested in the y coordinate of that lowest point.
The equation y = x^2+3x-4 is of the form y = ax^2+bx+c
where,
Note: this c value of -4 has nothing to do with the c value mentioned in the original inequality. It's an unfortunate clash of variable names.
Plug 'a' and b into the formula below to find the x coordinate of the vertex.
h = -b/(2a)
h = -3/(2*1)
h = -1.5
Then use this x value to find its paired y value
y = x^2+3x-4
y = (-1.5)^2+3(-1.5)-4
y = -6.25
The vertex is located at (-1.5, -6.25) which is the lowest point on the parabola.
The smallest y can get is -6.25
This is the same as saying the smallest (x-1)(x+4) can get is -6.25
If c from the original inequality is smaller than -6.25, then (x-1)(x+4) > c will always be true for all real numbers x.
Therefore, the interval of possible values of c is c < -6.25