Inequality rational polymonial

In summary: The Attempt at a SolutionThe attempted solution set is all values of x less than -4 and all values of x greater than 3/2.
  • #1
xzi86
9
0

Homework Statement


(3x+1)/(x+4)>=1


Homework Equations





The Attempt at a Solution



(3x+1)>=(x+4)
2x>=3
x>=3/2

But this is wrong?? Why?
 
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  • #2
If you multiply both sides of an inequality by a negative number, you reverse the direction of the inequality. But you don't know whether x+ 4 is positive or negative.

Instead break it into two parts. x+ 4 is positive, that is x+ 4> 0 for x> -4. If that is the case, then 3x+1> x+ 4 so that 2x> 3, x> 3/2. Because any value of x that is larger than 3/2 certainly is larger than -4, x>= 3/2 is part of the solution.

But if x< -4, then x+ 4 is negative. And then 3x+1< x+ 4 so that 2x< 3 so that x< 3/2. Now we argue the other way: any number that is less than -4 certainly is less than 3/2 so x< -4 is the other part of the solution. (x= -4 is not because the fraction is not defined if x= -4.)
You can check: if x= -5, then 3x+1= -15+1= -14, x+ 4= -1 so that (3x+1)/(x+4)= -14/-1= 14 which is certainly greater than 1.

That is, the solution set is all values of x less than -4 and all values of x greater than 3/2.
 
  • #3
so my method was good except i forgot to consider x is negative
 
  • #4
xzi86 said:
so my method was good except i forgot to consider x is negative

Not x is negative, but x+4 is negative, or x+4<0, x<-4.

Another way to solve this problem without considering cases is to multiply by (x+4)2 because any number squared is always positive, so there is no need to switch the inequality sign. You have to solve a quadratic and you'll get the same answer.
 
  • #5

What is "inequality rational polynomial"?

"Inequality rational polynomial" refers to an algebraic expression that contains rational coefficients and is set to be either greater than or less than a certain value.

How is "inequality rational polynomial" different from other polynomials?

The main difference between "inequality rational polynomial" and other polynomials is that the inequality symbol is used in the expression, rather than an equal sign. This indicates that the inequality rational polynomial has a range of possible solutions, rather than a single solution.

What is the purpose of studying "inequality rational polynomial"?

The study of "inequality rational polynomial" is important in various fields, such as economics, physics, and engineering. It allows for the analysis of real-life situations and models, where values are not always exact and can vary within a range.

What are the common methods for solving "inequality rational polynomial" equations?

There are several methods for solving "inequality rational polynomial" equations, including factoring, graphing, and using the quadratic formula. The most appropriate method may vary depending on the complexity of the equation and the desired level of accuracy.

How can "inequality rational polynomial" equations be applied in real-life situations?

"Inequality rational polynomial" equations can be applied in various real-life situations, such as calculating profit margins in business, predicting population growth in biology, and determining the optimal trajectory of a projectile in physics. They allow for the analysis of complex systems and can help make informed decisions.

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