Introduction to Calc inequality

Question

The inequality is x⁷ - x⁶ <= 0. So far, I think the first step is to factor out x^6, leaving us with (x^6)(x-1) <= 0, though I'm not sure about this step. Any help is appreciated!

Answer 1

When you factor, you get

 

x⁶(x - 1) = 0

 

 

This gives you the zeros    x = 0     and      x = 1.  The zeros are your reference points.

 

Next, I like to think of the polynomial graphically and using test points.  The degree of the polynomial is 7, so the continuous curve is an asymmetrical curve with that curve decreasing as x decreases infinitely, and curve increasing as x increases infinity.  Next, you want your curve to be below or on the x-axis.   Using test points, the values should be either negative or zero.   The interval in which this happens is

 

(-∞, 1]

Answer 2

x⁷ - x⁶ ≤ 0

 

x⁶(x - 1) ≤ 0

 

The left side of the inequality is equal to zero when x = 0 or 1.

 

Divide the number line into 3 intervals:  (-∞, 0), (0,1), and (1,∞)

 

On the interval (-∞,0), x⁶(x-1) < 0

 

On the interval (0,1), x⁶(x-1) < 0

 

On the interval (1,∞), x⁶(x-1) > 0

 

Solution:  (-∞, 1]

 

Answer 3

You're right. That's a good first step. factor out the x⁶

 

x⁶(x-1) ≤ 0

 

divide by x⁶ is just zero

which leaves

 

x - 1 ≤ 0

 

add 1 to both sides

 

x ≤ 1