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(i) Find the range of values of c for which (x - 1)(x + 4) >c for all real values of x.​

User Spenthil
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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,

  • a = 1
  • b = 3
  • c = -4

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

User Agrinh
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