MHB Can graph be used to solve inequalities without algebra?

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The discussion focuses on using the graph of the polynomial y = x^4 - 4x^3 + 6x^2 - 4x + 2 to solve the inequalities x^4 - 4x^3 + 6x^2 - 4x + 2 < 0 and x^4 - 4x^3 + 6x^2 - 4x + 2 > 0 without algebra. The graph indicates that it crosses the y-axis at one point and does not touch or cross the x-axis, suggesting that the polynomial is always positive. Therefore, the inequality x^4 - 4x^3 + 6x^2 - 4x + 2 > 0 is satisfied for all x, while x^4 - 4x^3 + 6x^2 - 4x + 2 < 0 has no solutions. The graph effectively illustrates the behavior of the polynomial in relation to the inequalities.
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How do we use the graph to solve a given inequality.

For example, say the graph of y = x^4 - 4x^3 + 6x^2 - 4x + 2 is given. The graph of y crosses the y-axis at one point. It does not touch or cross the x-axis. In what way can the picture, the graph help us solve either of the following inequalities given below?

A. x^4 - 4x^3 + 6x^2 - 4x + 2 < 0

B. x^4 - 4x^3 + 6x^2 - 4x + 2 > 0

Remember, use the graph to solve. Do not solve algebraically. How is this done?
 
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RTCNTC said:
How do we use the graph to solve a given inequality.

For example, say the graph of y = x^4 - 4x^3 + 6x^2 - 4x + 2 is given. The graph of y crosses the y-axis at one point. It does not touch or cross the x-axis. In what way can the picture, the graph help us solve either of the following inequalities given below?

A. x^4 - 4x^3 + 6x^2 - 4x + 2 < 0

B. x^4 - 4x^3 + 6x^2 - 4x + 2 > 0

Remember, use the graph to solve. Do not solve algebraically. How is this done?

Let's look at the graph as advised:

[DESMOS=-2.007008393275925,5.5746982450828035,-0.18167978042326505,2.4485740403429896]y=x^4-4x^3+6x^2-4x+2[/DESMOS]

Based on this, what values of $x$ appear to satisfy the two given inequalities?

By the way, do you recognize that:

$$y=(x-1)^4+1$$
 
The values of x? The graph does not touch the x-axis. It goes through one point on the line x = 0. I think 0 satisfies the inequality that is > 0 but does not satisfy the inequality where 0 is greater than the polynomial.
 
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