Solve Algebra Inequality: $\frac 1{3x^2-x-2}<0$

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Discussion Overview

The discussion revolves around solving the algebraic inequality $\frac{1}{3x^2-x-2}<0$. Participants explore the conditions under which the inequality holds, including the need to factor the denominator and analyze critical points.

Discussion Character

  • Mathematical reasoning
  • Debate/contested
  • Homework-related

Main Points Raised

  • Some participants express confusion about how to find the set of values satisfying the inequality and mention the requirement to make one side zero.
  • Questions are raised regarding the conditions for $\frac{1}{x}<0$ and how to approach solving the inequality.
  • One participant suggests factoring the denominator and checking intermediate values to determine where the function changes signs.
  • Another participant proposes using the quadratic formula to factor the denominator and identifies critical points at $x = -\frac{2}{3}$ and $x = 1$.
  • It is noted that for the inequality to hold, exactly one of the binomials in the factored form must be positive while the other is negative.
  • Participants discuss checking points in different intervals to confirm where the inequality is satisfied.

Areas of Agreement / Disagreement

There is no consensus on the solution process, as participants present varying approaches and interpretations of the inequality. Some agree on the critical points but differ on the method of checking intervals.

Contextual Notes

Participants mention the need to consider the continuity of the function and the implications of the denominator being undefined at certain points. There are unresolved steps regarding the specific intervals that satisfy the inequality.

Who May Find This Useful

This discussion may be useful for students learning about algebraic inequalities, particularly those seeking to understand the process of solving such inequalities and the importance of critical points and sign analysis.

Air
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\frac 1{3x^2-x-2}&lt;0

I understand that in algebra inequalities, one side has to be made zero but in the equation shown above, I am unable to find the set of values that satisfies the equation.
 
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For what values of x is 1/x < 0 ?
 
one side needs to be "made zero" for a continuous function to change signs, This function isn’t continuous where it’s undefined: I suggest you factor the bottom and check the intermediate values
 
genneth said:
For what values of x is 1/x < 0 ?

Yes, that is what needs to be worked out. I know the answer but when I do the working it doesn't come to the answer so I want to know how to do working out.

The answer is -2/3 < x < 1[/color].
 
Air said:
Yes, that is what needs to be worked out. I know the answer but when I do the working it doesn't come to the answer so I want to know how to do working out.

The answer is -2/3 < x < 1[/color].

No -- I mean if I gave you the (in)equation 1/x < 0, what's the solution in terms of x? Then I was going to ask you to solve x^2 - 1 < 0, followed by (x-a)(x-b) < 0. Then hoping that you'd managed the rest...
 
Use the quadratic formula to factorize the denominator.
 
Air said:
\frac 1{3x^2-x-2}&lt;0

I understand that in algebra inequalities, one side has to be made zero but in the equation shown above, I am unable to find the set of values that satisfies the equation.
The denominator factors to two binomials. In order for the original rational left side to be less than zero, exactly ONE of the binomials must equal zero but not both equal to zero.

The binomials for the denominator are (3x+2) and (x-1). Additionally, x must NOT equal -2/3, and x must NOT equal +1.

Critical points to use for boundary between solution ranges might be x at -2/3 and x at +1.
 
... in fact, the solution appears to be disjoint , but actually checking each section of the number line tells you something more specific.

Check a point less than -2/3;
Check a point more than -2/3 and less than +1;
check a point more than +1 (should not really be necessary).

The values of x between -2/3 and +1 exclusive should be the solution.
 
Air said:
\frac 1{3x^2-x-2}&lt;0

I understand that in algebra inequalities, one side has to be made zero but in the equation shown above, I am unable to find the set of values that satisfies the equation.
\frac 1{3x^2-x-2}&lt;0

\Rightarrow3x^2-x-2&lt;0

\Rightarrow(3x+2)(x-1)&lt;0

i think it's easy from here.
 
  • #10
from murshid_islam
\Rightarrow(3x+2)(x-1)&lt;0

With that, you need just to look at two situations:
3x+2<0 AND x-1>0
OR (which you must check carefully to see whether conjoint or disjoint)
3x+1>0 AND x-1<0

Important is that one binomial must be positive and the other binomial must be negative.
 

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