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

Question

Solve each rational inequality. Give the solution set in interval notation. See Examples 8 and 9. 4/(x+1)<2/(x+3)

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 X plus four is less than two divided by the quantity of X plus seven. Now our first step in solving this inequality is to get all of our terms on the same side of the inequality. So we can begin by subtracting two divided by the quantity of X-plus seven by both sides of our inequality. So now rewriting our inequality, we have five divided by the quantity of X plus four minus two divided by the quantity of X plus seven. And this is now less than zero. So now our next step is to combine both of our fractions so that we can simplify the left side of our inequality. So to combine the fractions, we need to get a common denominator. So we want to multiply our first fraction by the expression in the denominator of our second fraction. So five divided by the quantity of X plus four gets multiplied by the quantity of X plus seven divided by the quantity of X plus seven. And then our second fraction gets multiplied by the expression in the denominator of the first fraction. So two divided by the quantity of X plus seven gets multiplied by the quantity of X plus four divided by the quantity of X plus four. And now we just need to simplify this expression. So first, we have five times the quantity of X plus seven all divided by the quantity of X plus four times the quantity of X plus seven. And now our second fraction, we will then have minus two times the quantity of X plus four all divided by the quantity of X plus four times the quantity of X plus seven. And this whole expression is still just less than zero. So now we want to simplify our left side of our inequality further by combining our numerator since we have common denominators. So combining the enumerators, we first need to factor in the five and the negative two. So we get five X plus 35 five minus two, X minus eight all divided by the quantity of X plus four times the quantity of X plus seven. And this expression is still less than zero. Now, we just need to simplify our numerator by combining the like terms. And so our simplified inequality becomes the quantity of three X plus 27 divided by the quantity of X plus four times the quantity of X plus seven. And this expression is still less than zero. So now with this simplified inequality, we want to start finding X values by setting the numerator and the denominator individually equal to zero. So beginning with our numerator, we have three X plus 27. Again, we set this expression equal to zero. And now solving for X, we first need to subtract 27 from both sides and we get three X is equal to negative 27. And now we need to get rid of the coefficient of X by dividing both sides by three. Now, simplifying we see that one of our solutions for X is negative nine. And now we need to repeat this process with the denominator from our inequality. So we have the quantity of X plus four times the quantity of X plus seven. And we need to set this expression equal to zero. Now solving for X, we can actually set each individual part of the expression equal to zero. So we can begin with X plus four, setting it equal to zero. We now can solve for X by subtracting four from both sides. And we see that we get one solution as X being negative four and that we can repeat this with the second part of our expression where we had X plus seven. And now we set this equal to zero. Solving for X, we subtract seven from both sides telling us a third solution for X is negative seven. So now with these solutions. These values that we found for X, we can use them to write intervals that describe the X axis. So we see the X axis can be separated into the intervals where we go first from negative infinity to the smallest X value which is negative nine. Then we close both of these terms by parentheses. And then we go from negative nine to the next smallest solution which is negative seven. And again, both of these values are closed by parentheses. Then we go from negative seven to the next solution which is negative four. And both of these terms are closed parenthesis. And lastly, we go from the last solution or the largest solution of negative four to positive infinity. And now with these four intervals, we need to test each of them to see which are a solution to our inequality, our simplified inequality from earlier. So to do this, we first need to let F of X be the left side of our inequality where we have the quantity of three X plus all divided by the quantity of X plus four times the quantity of X plus seven. So now to test our intervals, we need to set up a table. So first, we will take one of our intervals and then we will choose a test value that is within the interval. And then with that test value, we need to substitute it into our function F of X and from the results of our function depending on the sign of the value, we can form a conclusion about F of X. So we can begin with our first interval where we go from negative infinity to negative nine. And now we just need to choose a test value within this interval and we can choose negative 10. So now we substitute negative 10 into F of X. So F of negative 10 gives us three times negative 10 plus 27 all divided by the quantity of negative 10 plus four times the quantity of negative 10 plus seven. And now simplifying, we see that F of negative 10 gives us negative 16. So since we obtained a negative value from our function, we can conclude that F of X is going to be less than zero for all X values within the interval from negative infinity to negative nine. Now we can move on to testing our next interval where we go from negative nine to negative seven. Now we need to input a test value that is within the interval. So we can choose negative eight. Now we need to find F of negative eight, F of negative eight gives us three times negative eight plus 27 all divided by the quantity of negative eight plus four times the quantity of negative eight plus seven. Now simplifying, we see we get 17 divided by 60. Now, since we obtained a positive value, we can conclude that F of X is going to be greater than zero for all X values within the interval from negative 92, negative seven. Now we have tested our second interval so we can move on to the next interval where we go from negative seven to negative four. And now choosing a test value here, we can just choose negative five. So now we substitute in negative five into F of X. So finding F of negative five, we have three times negative five plus 27 all divided by the quantity of negative five plus four times the quantity of negative five plus seven. Now simplifying, we see that from negative five, we get negative six. And since we obtained a negative value, we can conclude that F of X is going to be less than zero for all X values from or within the interval from negative seven to negative four. And now we can test our last interval where we go from negative four to positive infinity. We can choose a test value within this interval as just zero. Now finding F of zero, we have three times zero plus 27 all divided by the quantity of zero plus four times the quantity of zero plus seven. Simplifying we get 27 divided by 28. And we see that this is a positive value. So we can conclude that F of X will be greater than zero for all X values within the interval from negative four to positive infinity. So now taking a look at all of our conclusions that we found for each interval, we can conclude that of X is going to be less than zero for all X values for all X values from negative infinity to negative nine and from negative seven to negative four. So here we can see that these two intervals cause our expression to be less than zero, which is the solution for our inequality because we were looking for our function to be less than zero. So these two intervals are our final solution. And this then leaves us 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/(x+1)<2/(x+3)

Key Concepts

Rational Inequalities

Finding a Common Denominator and Combining Fractions

Sign Analysis and Interval Testing

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