Solving Tricky Inequalities: Help with a Non-Factorable Numerator

  • Context: High School 
  • Thread starter Thread starter J-Girl
  • Start date Start date
  • Tags Tags
    Inequalities
Click For Summary

Discussion Overview

The discussion revolves around solving a tricky inequality involving a non-factorable numerator. Participants explore methods for addressing the inequality (x^2-4x+7)/(x^2+x-6)≤0, focusing on the challenges posed by the numerator's inability to be factored.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Conceptual clarification

Main Points Raised

  • One participant expresses difficulty in solving the inequality due to the numerator not being factorable.
  • Another suggests rewriting the numerator as (x-2)^2+3 and questions the implications of this form.
  • A participant clarifies that the inequality may involve a specific form, such as \frac{x^2-4x+7}{x^2+x-6}>1, and proposes multiplying both sides by (x^2+x-6)^2, emphasizing not to expand.
  • The original poster later clarifies that the inequality is (x^2-4x+7)/(x^2+x-6)≤0 and expresses uncertainty about multiplying by the denominator without knowing its sign.
  • Another participant prompts the original poster to consider when (x-2)^2+3 is positive or negative, highlighting the nature of the squared term.
  • The original poster indicates a breakthrough in understanding after the discussion.

Areas of Agreement / Disagreement

The discussion reflects a lack of consensus on the best approach to solve the inequality, with multiple perspectives on how to handle the numerator and denominator. Participants engage in clarifying questions and suggestions without reaching a definitive solution.

Contextual Notes

There are limitations regarding the assumptions about the positivity or negativity of the denominator, as well as the implications of the rewritten form of the numerator. The discussion does not resolve these aspects.

J-Girl
Messages
38
Reaction score
0
Inequalities- tricky question!

hiii,
i was wondering if anybody knew how to help me with this one tricky homework question. i can do most of the inequaliies I've come across, but how do you solve an inequality if you can't factor it?
the question is: (x^2-4x+7)/(x^2+x-6)
i know that the denominator is easily factorable ((x+3)(x-2)) but the numerator isnt? help pleasezzz I've been stuck on this for ages!:(:(
 
Mathematics news on Phys.org


Rewrite (x^2-4x+7) to (x-2)^2+3. What can you say about this?
 


Ok so I'm guessing by inequalities you mean something like [tex]\frac{x^2-4x+7}{x^2+x-6}>1[/tex] for example?

Try multiplying both sides by [tex](x^2+x-6)^2[/tex] since you know this has to be a non-negative number, and don't expand! Use your knowledge of factorizing to solve it.
 


ohh sorry i didnt even put the rest of the question in! it was (x^2-4x+7)/(x^2+x-6)≤ 0
i just wasnt sure if i was allowed to automatically multiply by the denominator because i didnt know if it was positive or negative. if i rephrase it as (x-2)^2 + 3, i still can't cancel out any brackets because its not completely factored. sorry:( just haven't done maths in ages and i suck at it..
 


Can you tell when (x-2)^2+3 is positive and negative though? Notice the squared term.
 


oohhh i got it now!:) yay took ages tho lol but thanks every1:)
 

Similar threads

MHB How Do You Solve and Graph This Quadratic Inequality?
  • · Replies 2 ·
Replies
2
Views
2K
MHB How do we solve a quadratic inequality with multiple factors?
  • · Replies 3 ·
Replies
3
Views
2K
MHB Can graph be used to solve inequalities without algebra?
  • · Replies 2 ·
Replies
2
Views
2K
MHB Inequality with fractions solve 2/(x^2−1)≤1/(x+1)
  • · Replies 9 ·
Replies
9
Views
2K
MHB What is the Factorization and Inequality of a Lagoon's Depth Function?
  • · Replies 7 ·
Replies
7
Views
3K
Need help in manipulating rational absolute value inequalities
  • · Replies 11 ·
Replies
11
Views
3K
MHB Solving Inequality: 0 <= 6x3 - 24x2 + 6x - 10
  • · Replies 2 ·
Replies
2
Views
2K
High School Solve Linear Inequality: ABS Value(7x-8) <=4x+7
  • · Replies 4 ·
Replies
4
Views
2K
MHB How do I solve a quadratic inequality using factoring?
  • · Replies 7 ·
Replies
7
Views
2K
High School Why Can't We Cancel (x-2) in the Inequality 2(x+3)(x-2)/(x-2)≥0?
  • · Replies 8 ·
Replies
8
Views
2K