- #1
Achi_kun
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Rational Inequalities: Solve & Understand | Math
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In summary, the conversation is about solving the inequality $\dfrac{1}{x} > 2$ by multiplying on both sides by x and considering two cases, x>0 and x<0. The solution is $0<x<\dfrac{1}{2}$, and it is also mentioned that $\dfrac{1}{x}$ is undefined for x=0. To change from f(x)<g(x) to f(x)>g(x), the equation $\dfrac{1}{x}=2$ needs to be solved and three intervals, x<0, 0<x<1/2, and x>1/2, need to be considered. Ultimately, the summary explains that every number between
- #2
skeeter
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Uhh ... fill in the blanks? Have you worked on any of these?
- #3
Achi_kun
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Idk how the solution worksskeeter said:
- #4
skeeter
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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
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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 betwen 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 betwen 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.
FAQ: Rational Inequalities: Solve & Understand | Math
What is a rational inequality?
A rational inequality is an inequality that involves one or more rational expressions. A rational expression is a fraction in which the numerator and denominator are polynomials.
How do I solve a rational inequality?
To solve a rational inequality, follow these steps:
- 1. Move all terms to one side of the inequality so that the right side is equal to 0.
- 2. Factor the numerator and denominator of each rational expression.
- 3. Determine the critical values by setting each factor equal to 0 and solving for the variable.
- 4. Use a sign chart to determine the intervals where the inequality is true.
- 5. Check the intervals using test points to determine the final solution.
What are critical values in a rational inequality?
Critical values are the values of the variable that make the denominator of a rational expression equal to 0. These values are important because they determine the intervals where the inequality may change from true to false or vice versa.
Can a rational inequality have more than one solution?
Yes, a rational inequality can have more than one solution. This is because the solution to a rational inequality is a set of intervals, rather than a single value. These intervals can include multiple values that satisfy the inequality.
How can I check my solution to a rational inequality?
You can check your solution to a rational inequality by plugging in test points within each interval and determining if the inequality is true or false. If the inequality is true for all test points in an interval, then that interval is part of the solution. If the inequality is false for any test point in an interval, then that interval is not part of the solution.