Solve each rational inequality. Give the solution set in inter...

Question

Solve each rational inequality. Give the solution set in interval notation. See Examples 4 and 5.

$\frac{5}{x + 3} \geq \frac{3}{x}$

Accepted Answer

Provide the solution set for the given rational inequality and interval notation where we have six divided by X plus 17 is square, there equals to four divided by X to solve this. We want all of our rational expressions onto one side. Let's first subtract by four divided by X. We then get six divided by X plus 17 minus four divided by X is greater or equals to zero. Now, we want to combine these two expressions, go ahead and find a common denominator to do that multiply our first fraction by X divided by X and our second one by X plus 17 divided by X plus 17. This gets a six X divided by X plus minus four multiplied by X plus 17, divided by X plus 17 multiplied by X. We also have a multiplication by X on our first fraction. Now, we can combine by multiplying over R4, you will get 6 x divided by X plus 17, multiplied by X minus four, X minus divided by X plus 17 multiplied by X. This simplifies to two X minus 68 divided by X plus 17, multiplied by X. We can divide by two on both sides of this inequality. You get X minus 34 divided by X plus 17, multiplied by X is greater and equals to zero. We cannot find our discontinuities to do that. We will set our numerator equals to zero X minus 34 S equal to zero, X for 17 is equal to zero and X is equal to zero. We then get X is equals 34 x equals negative 17. We can then set up our intervals. Our intervals will be from native infinity to native 17, -17-0, 0 - 34 and 34 to Infinity. No, we want to find values on these intervals. That will give us a specific sign. In this case, we want to be greater in or equals to zero. So we want a positive sign on these intervals. Let's test some values on each interval to determine what our sign will be. For the first one. Let's say we check X is 18 or negative 18. We would get negative 18 minus 34 divided by negative plus 17, multiplied by negative 18. That results to -26 divided by nine. That interval is negative. Let's say you want f on the next interval, let's say we use the value -1. If we were to plug that in, you would get 35 divided by 16. That is positive. Let's take another value on our next interval. I'll check half of one when X is equal to one, we plugged in the one, we would get negative 33 divided by 18. That is negative. Finally, we, I can value on our last interval. Let's check 35. We plugged in 35. You would get one divided by 1820. That is positive. So we can determine then our positive intervals, our native 17-0 M 34 to infinity, which means the answer to our problem will be the intervals negative of 17- And 34 to Infinity. Now, one last thing you wanna check is our brackets. We have a called Susan. So that means we have a square racket somewhere because the answers for -170 were in the denominator of our expression. We will leave those as round brackets. 34 will be our square bracket. So our answer, it's actually 34 to infinity. For the upper interval with a square bracket. We can then conclude the answer to our problem has answer a OK. I hope to help you solve the problem. Thank you for watching. Goodbye.

Solve each rational inequality. Give the solution set in interval notation. See Examples 4 and 5.
$\frac{5}{x + 3} \geq \frac{3}{x}$

Key Concepts

Rational Inequalities

Finding a Common Denominator and Combining Terms

Sign Analysis and Interval Notation

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