Stuck on chapter polynomial and rational inequalities
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blamocur
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Start with polynomials before moving to rational functions.
Where are you stuck in your book?
Do you have a problem with specific definition, theorem or statement?
Can you solve [imath]x^2-1 \leq 0[/imath] ?
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Dr.Peterson
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Here is an online textbook section on the topic:
3.8: Polynomial and Rational Inequalities
Solve polynomial and rational inequalities by using a sign chart.
math.libretexts.org
You may want to consider this or similar online resources as part of your study. Here is another that I have long recommended:
Algebra - Polynomial Inequalities
In this section we will continue solving inequalities. However, in this section we move away from linear inequalities and move on to solving inequalities that involve polynomials of degree at least 2.
tutorial.math.lamar.edu
Steven G
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Can you please post your solution as my students get this problem wrong.
Otis
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Hi bladeren. Are you asking about the form of your answers? That is, expressing intervals by writing square brackets and/or parentheses on a labeled number line sketch? Or, are you asking about using a number line to organize your work?
In the former case, the only reason you "must" write an answer that way is when you're instructed to. Otherwise, you may use set builder notation or a compound inequality.
In the latter case, you're probably free to organize your work in some other way (eg: a chart), as long as it's clear. I think the number line serves as a good visual reference because it eliminates the need to manipulate a lot of information in your head at once.
If you're asking about some other use, please clarify. Cheers
[imath]\;[/imath]
You have a little mistakes. The critical numbers are [imath]-1[/imath] and [imath]1[/imath] and you want to know what are the signs between them.
[imath](-2 - 1)(-2 + 1) = +[/imath]
[imath](0 - 1)(0 + 1) = -[/imath]
[imath](2 - 1)(2 + 1) = +[/imath]
What does this tell you if we need to satisfy the condition [imath](x - 1)(x + 1) \leq 0[/imath]?
Which one of the three above is less than zero?
Less than zero means negative.
[imath](-2 - 1)(-2 + 1) = +[/imath]
[imath](0 - 1)(0 + 1) = -[/imath]
[imath](2 - 1)(2 + 1) = +[/imath]
Which one of them is negative? Look on the right side!
This means [imath]x[/imath] is between [imath]-1[/imath] and [imath]1[/imath]. Therefore the domain of [imath]x[/imath] is:
[imath]-1 \leq x \leq 1[/imath]
It's not about the book is good or bad, it's about this example [imath]x^2 - 1 \leq 0[/imath] which is basically difficult even for students who are good at mathematics because it is tricky! Professor blamocur has given you this example because he knows 99% of the students will solve it wrong. Professor Steven emphasized on it too because his students have already solved it wrong!
If you really understood this example, you can solve all inequality questions safely now because this was the most difficult one! (At least for me.)
The alternative form for [imath]x^2 - 1 \leq 0[/imath] is:
[imath]|x| \leq 1[/imath]
And remember this absolute value, it will always make your life difficult at math even at the highest level of mathematics.
Last edited:
Dr.Peterson
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You tried the number [imath]x = -3[/imath], and found that the factor [imath](x+4)[/imath] is positive and the factor [imath](x-1)[/imath] is negative, so their product is negative:
(You could also have found directly that [imath](-3)^2+3(-3)-4=0-9-4=-4[/imath] is negative, though that takes more work.)
Since the sign of the expression can only change at critical points, this shows that it will be negative throughout the interval (-4,1). Therefore, the inequality (which says it's positive) is not true in that interval. It is true in the other two intervals.