- #1
Achi_kun
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Rational Inequalities: Solve & Understand | Math
- Context: MHB
- Thread starter Achi_kun
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- Inequalities Rational
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Discussion Overview
The discussion revolves around solving and understanding rational inequalities, specifically the inequality $\dfrac{1}{x} > 2$. Participants explore different methods for solving the inequality, including considerations of cases based on the sign of x and the implications of multiplying by x.
Discussion Character
- Exploratory
- Mathematical reasoning
- Debate/contested
Main Points Raised
- Some participants express confusion about the solution process for the inequality.
- One participant proposes a method involving combining fractions to manipulate the inequality.
- Another participant suggests a different approach by multiplying both sides by x, emphasizing the need to consider cases based on whether x is positive or negative.
- In the positive case, the participant concludes that $0 < x < \frac{1}{2}$ satisfies the inequality.
- In the negative case, they argue that no values satisfy the inequality since it leads to a contradiction.
- Further, the participant discusses the importance of identifying points where the functions are equal or undefined to determine intervals for testing values.
- They provide specific test values for each interval to verify which satisfy the inequality.
Areas of Agreement / Disagreement
Participants do not reach a consensus on the best method for solving the inequality, with differing approaches and interpretations of the steps involved. Some express uncertainty about the solution process.
Contextual Notes
Participants highlight the need for careful consideration of the sign of x when manipulating the inequality, as well as the importance of identifying critical points where the function is undefined or equal.
- #2
skeeter
- 1,103
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Uhh ... fill in the blanks? Have you worked on any of these?
- #3
Achi_kun
- 5
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Idk how the solution worksskeeter said:
- #4
skeeter
- 1,103
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for $\dfrac{1}{x} > 2$
step 1. $\dfrac{1}{x} - \dfrac{2}{1} > 0$
combine the two fractions by using a common denominator
step 1. $\dfrac{1}{x} - \dfrac{2}{1} > 0$
combine the two fractions by using a common denominator
- #5
HOI
- 921
- 2
Personally, I wouldn't do it that say.
From $\frac{1}{x}> 2%$, multiply on both sides by x.
But you have to be careful with that! Unlike with an equation, multiplying on both sides by negative number reverses the ">" sign. So do two cases:
1) If x> 0 then $1>2x$. Divide on both sides by the positive number 2: $\frac{1}{2}> x$..
Since we are requiring that x be positve, we have $0< x< \frac{1}{2}$
2) If x< 0 then $1< 2x$. Divide on both sides by the positive number 2: $\frac{1}{2}< x$, But since we are requiring that x be negative, that is not possible.
The solution is $0<x \frac{1}{2}$.
It is also true that, for continuous functions, g and f, to change from f(x)<g(x) to f(x)> g(x), we have to go through f(x)= g(x) or a poinr where either f or g is undefined.
So start by solving the equation $\frac{1}{x}= 2$. That is the same as $1= 2x$, or $x=\frac{1}{2}$. I is also true that $\frac{1}{x}$ is undefined for x= 0. That divides the real numbers into three intervals, x< 0, 0< x< 1/2, and x> 1/2. We need only check one value of x in each interval. For x< 0 take x=-1. Then 1/x= -1 which is NOT larger than 2 so no x less than 0 satisfies 1/x> 2. For 0< x< 1/2 we can take x=1/4. Then 1/x= 4 which is greater than 2. Every number between 0 and 1/2 satisfies the inequalty. Finally take x= 1. Then 1/x=1 which is not larger than 2. No x larger than 1/2 satisfies the inequality.
From $\frac{1}{x}> 2%$, multiply on both sides by x.
But you have to be careful with that! Unlike with an equation, multiplying on both sides by negative number reverses the ">" sign. So do two cases:
1) If x> 0 then $1>2x$. Divide on both sides by the positive number 2: $\frac{1}{2}> x$..
Since we are requiring that x be positve, we have $0< x< \frac{1}{2}$
2) If x< 0 then $1< 2x$. Divide on both sides by the positive number 2: $\frac{1}{2}< x$, But since we are requiring that x be negative, that is not possible.
The solution is $0<x \frac{1}{2}$.
It is also true that, for continuous functions, g and f, to change from f(x)<g(x) to f(x)> g(x), we have to go through f(x)= g(x) or a poinr where either f or g is undefined.
So start by solving the equation $\frac{1}{x}= 2$. That is the same as $1= 2x$, or $x=\frac{1}{2}$. I is also true that $\frac{1}{x}$ is undefined for x= 0. That divides the real numbers into three intervals, x< 0, 0< x< 1/2, and x> 1/2. We need only check one value of x in each interval. For x< 0 take x=-1. Then 1/x= -1 which is NOT larger than 2 so no x less than 0 satisfies 1/x> 2. For 0< x< 1/2 we can take x=1/4. Then 1/x= 4 which is greater than 2. Every number between 0 and 1/2 satisfies the inequalty. Finally take x= 1. Then 1/x=1 which is not larger than 2. No x larger than 1/2 satisfies the inequality.
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