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. (x+3)/(2x-5)≤1

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 where we have the quantity of X plus 12 divided by the quantity of four X minus nine is less than or equal to one. Now, our first step in solving this inequality is to, to get all of our terms on the same side of the inequality. So we just want to begin by subtracting one from both sides. So our inequality becomes the quantity of X plus 12 divided by the quantity of four X minus nine minus one is less than or equal to zero. So now our next step is to simplify the left side of our inequality by combining our whole number with the fraction. So we need to get a common denominator between the whole number and the fraction. So we just need to multiply one by the expression in the denominator. So one gets multiplied by the quantity of four X minus nine divided by the quantity of four X minus nine. Now we just need to simplify this expression. And so our inequality becomes the quantity of X plus divided by the quantity of four X minus nine minus one times the quantity of four X minus nine all divided by the quantity of four X minus nine. And again, this expression is still less than or equal to zero. So now that we have found common denominators between both of our terms, we just need to simplify the left side further by combining the enumerators. So doing this, we have X plus 12 and distributing the negative one, we have minus four X plus nine all divided by the quantity of four X minus nine. And this expression is less than or equal to zero. So now our next step in simplifying is to just combine the like terms in the numerator. So doing this, our simplified inequality become the quantity of negative three X plus 21 divided by the quantity of four X minus nine is less than or equal to zero. So now we want to find values for X and we need to do this by setting the numerator and the denominator individually equal to zero. So we can begin with the numerator where we have negative three X plus 21. And again, we set this expression equal to zero. So we need to solve for X by first subtracting 21 from both sides. And this gives us negative three X is equal to negative 21. And now we need to get rid of the coefficient. So we need to divide both sides by negative three. And this gives us one solution for X as positive seven. So now our next step is to repeat this process with the denominator. So we have four X minus nine and we need to set this equal to zero. Now solving for X, we first add nine to both sides which gives us four, X is equal to nine. And then we need to get rid of the coefficient by dividing both sides by four. And this gives us our second value for X which is nine fourth. And now we can take these two solutions that we found for X and write them as intervals. So we see that the X axis can be separated into the following intervals where we go first from negative infinity to the smallest solution that we found, which is 9/4. And both of these values are closed by parentheses for now. And then we go from the solution of nine and forth to the next solution which is seven. And again, both of these values are closed by parentheses. And lastly, we go from the largest solution which is seven all the way to positive infinity. And again, both of these terms are closed by parentheses. So now our next step is to test each of these intervals to see which of them serve as a solution to our inequality. So first, we need to let the function F of X be equal to the left side of our inequality where we have the quantity of negative three X plus 21 divided by the quantity of four X minus nine. So now with our function identified, we can begin testing hour intervals. So to test our intervals, we want to set up a table where we first take one of our intervals and then we choose a test value that is within the interval. And then we take this test value and we substitute it into our function F of X. And then with the value we get from F of X, we can form some conclusion about F of X based on the sign of the value. So we can begin with our first identified interval where we go from negative infinity to 9/4. Now we need to choose a test value within this interval. So we can choose zero. Now we need to find F of zero and substituting in zero into our function, we get negative three times zero plus 21 all divided by four times zero minus nine. And now simplifying, we see we get from F of zero, we get negative seven thirds. And now, since we obtained a negative value from our function, we know or can conclude that F of X is going to be less than zero for all X values within this interval from negative infinity to 9/4. Now we can move on to testing our next inter where we go from 9/4 to 7. And now we need to choose a test value in between this interval. So we can choose three. And now we need to substitute three into our function. So finding F of three, we get negative three times three plus 21 all divided by four times three minus nine. And now simplifying we get positive four. And since our result is positive, we can conclude that F of X will be greater than zero for all X values within the interval from 9/4 to 7. And now we can move on to our final interval where we go from seven to infinity and a test interval or test value within this interval, we can choose is just eight. So now finding F of eight, we have negative three times eight plus 21 all divided by four times eight minus nine. And now simplifying we get negative three divided by 23. So with this negative value, we see or can conclude that F of X will be less than zero for all X values within the interval from seven to positive infinity. So now, from each of our conclusions here, we see that F of X is going to be less than zero for all X values for all X values from the intervals of negative infinity to 9/ and from seven to positive infinity. So now we see again that these intervals are for when FFXF less than zero But if we recall from our inequality from our given inequality, we are looking for when F of X is less than or equal to zero. So that means we just need to adjust these intervals so that we include the values that cause F of X to be equal to zero. So F of X will be less than or equal to zero for all X values from negative infinity to 9/4. And now we know infinity is always excluded. So negative infinity gets closed by a parenthesis. But now 9/4 will also be closed by a parenthesis. Since 9/4 causes our denominator to be equal to zero. And while we don't want to divide by zero, so we need to exclude this value. And then for our next interval, we go from seven to infinity while positive infinity will also get closed by a parentheses since infinity is always excluded, but seven needs to be closed by a bear racket. Since seven causes our numerator to be equal to zero. Thus causing F of X to be equal to zero. So again, we need to include this value of seven and close it by a bracket. So with this, we have found our solution to our given inequality and written it in interval notation. And so with this, we are left here with answer choice. B 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. (x+3)/(2x-5)≤1

Key Concepts

Rational Inequalities

Critical Points and Sign Analysis

Interval Notation

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