Can graph be used to solve inequalities without algebra?

In summary, the graph of y = x^4 - 4x^3 + 6x^2 - 4x + 2 can help us solve inequalities by visually identifying which values of x satisfy the inequality. By looking at the graph, we can see that the inequality x^4 - 4x^3 + 6x^2 - 4x + 2 < 0 is satisfied by values of x that are less than 0, while the inequality x^4 - 4x^3 + 6x^2 - 4x + 2 > 0 is satisfied by values of x that are greater than 0. The graph does not touch the x-axis, but it crosses
  • #1
mathdad
1,283
1
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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  • #2
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:

\(\displaystyle y=(x-1)^4+1\)
 
  • #3
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.
 
Last edited:

FAQ: Can graph be used to solve inequalities without algebra?

What is a graph to solve inequality?

A graph to solve inequality is a visual representation of an inequality equation. It shows the values that satisfy the inequality and those that do not, by shading the region on the number line.

How do you read a graph to solve inequality?

To read a graph to solve inequality, look at the shaded region on the number line. The values in that region are the solutions to the inequality. The shaded region may be above or below a line, depending on the inequality symbol used.

What is the difference between a solid and a dashed line on a graph to solve inequality?

A solid line indicates that the value on the line is included in the solution, while a dashed line indicates that the value on the line is not included in the solution.

How do you graph a compound inequality?

To graph a compound inequality, first graph each inequality separately. Then, identify the overlapping region between the two graphs. This overlapping region represents the solutions to the compound inequality.

Can a graph to solve inequality have multiple solutions?

Yes, a graph to solve inequality can have multiple solutions. This is often the case with compound inequalities, where the overlapping region represents the solutions to the compound inequality.

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