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

Question

Solve each rational inequality. Give the solution set in interval notation. See Examples 8 and 9.
4/(3+x)≤3/(3+x)

Accepted Answer

Hey, everyone here we are asked to solve the following inequality and express the answer in interval notation. And here we are given the inequality five divided by the quantity of nine plus X is less than or equal to two divided by the quantity of nine plus X. Now our first step in solving this inequality is to get the right side to be zero. So we just need to subtract two divided by the quantity of nine plus X from both sides of our inequality. And so now rewriting our inequality, we have five divided by the quantity of nine plus X minus two divided by the quantity of nine plus X is less than or equal to zero. So now looking at the expression on the left side of our inequality, we see that both fractions have the same denominator. So we can simplify this expression further by combining their numerator. So combining the enumerators, we have the quantity of five minus two divided by the quantity of nine plus X and this is less than or equal to zero. So now we just need to simplify further by combining the like terms in the numerator. So five minus two is just three. So our simplified inequality is three divided by the quantity of nine plus X is less than or equal to zero. So now we have found a simplified form of our inequality where the right side of the inequality is just zero. So now we need to find solutions for X by setting our denominator equal to zero. So taking the expression in the denominator nine plus X, we set this expression equal to zero. So we can solve for X. And now we just need to subtract nine from both sides to get our X term alone. And we see our value for X is negative nine. And now with this solution for X, we know that the X axis can be separated into the following intervals where we first go from negative infinity to our solution of negative nine. And then we go from negative nine to positive infinity. And all of these terms for now are going to be close posed by parentheses. So now we need to test these two intervals that we came up with here and we test them by first letting F of X be equal to the left side of our simplified inequality where we have three divided by the quantity of nine plus X. And now with our function clearly identified, we can move on to testing the intervals. So to test the intervals, we need to set up a table where we first take the interval, we take the interval and then we choose a test value that's within this interval. And then with this test value, we substitute the value into our function F of X. And then with the value we get from F of X, we can form some sort of conclusion about our function. So we can begin with our first identified interval where we go from negative infinity to negative nine. And now we need to choose a test value within this interval here, we can choose negative 10. Now substituting this value into our function, we need to find F of negative 10, F of negative 10 gives us three divided by nine plus negative 10. Simplifying we get negative three. And now, since we obtained a negative value from our function, we can conclude that F of X is going to be negative or less than zero for all X values within the interval that we were testing, which is from negative infinity to negative nine. And now we can repeat this with our other interval where we go from negative nine to positive infinity. Now again, we choose a test value within this interval here, we can choose zero. Now finding F of zero, we have three divided by nine plus zero. And now simplifying we get three nights which is just one third. And since we obtained a positive value from our function, we can now conclude that F of X is going to be greater than zero or positive for all X values within the interval that we were testing, which is from negative nine to infinity. So now recalling our inequality from earlier, we were looking for when F of X is going to be less than or equal to zero. And now with this, less than sign, we need to look at our conclusions and find where F of X is just less than zero. And now we see from our conclusion that we get less than zero from our first interval. So F of X is going to be less than zero for all X values within the interval from negative infinity to negative nine. And now we need to adjust this interval so that it includes for when F of X is also equal to zero. So F of X is less than or equal to equal to zero for all X values within the interval from negative infinity to negative nine. Now looking at these two end values in our interval, we need to note that the new writer of F of X is a constant three. So there isn't going to be a value of X that will make F of X equal to zero. So the solution set is going to remain the same where we have the interval from negative infinity to -9. And both of these values are still going to be closed by parentheses. So now with this interval, we have found the solution set to our given inequality and interval notation. And so we are left with answer choice. C thanks for watching. I hope you found this video helpful and I'll see you again next time.

Solve each rational inequality. Give the solution set in interval notation. See Examples 8 and 9. 4/(3+x)≤3/(3+x)

Key Concepts

Rational Inequalities

Interval Notation

Critical Points

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