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

Question

Solve each rational inequality. Give the solution set in interval notation. (9x-11)(2x+7)/(3x-8)^3>0

Accepted Answer

Hey, everyone here we are asked to solve the following inequality and express the answer in interval notation. And we're given the inequality where we have the quantity of three X plus 17 times the quantity of two X minus five all divided by the quantity of two X minus seven. Cute. And now this whole expression is greater than zero. So now our first step is to find values of X and we do this by first setting our numerator equal to zero. So we take our numerator where we have the quantity of three X plus 17 times the quantity of two X minus five. And again, we need to set this equal to zero. So now we just need to solve for the values of X that make the left hand side expression equal to zero. So that means we can take the expression in each set of parentheses and individually set them equal to zero. So we can start with our first part of the expression where we have three X plus 17. Again, we need to set this equal to zero. And solving for X, we first need to subtract 17 from both sides. This gives us three, X is equal to negative 17. And now to get X alone, we divide both sides by three. And we see that one of our solutions for X is negative 17 3rd. So now we can take our second part of the expression where we have two X minus five. Again, we set this equal to zero. And solving for X, we first add five to both sides which gives us two X is equal to five. And now we divide both sides by two. And our next solution we have found for X is five halves. So now we repeat this process with the denominator of our original expression. So we take the denominator which is the quantity of two X minus seven cute and set it equal to zero. And now we can start by finding our values for X by expanding this expression on the left hand side. And so this cubed expression becomes the quantity of two X minus seven times the quantity of two X minus seven times the quantity of two X minus seven. And again, this is still equal to zero. And now, before we took each part of our expression and individually set it equal to zero. But here, since each part is, it is equal to each other, we just need to take one part of the expression. So we take two X minus seven and set it equal to zero. Now solving for X, we first add seven to both sides, which gives us two X is equal to seven. Now we divide both sides by two. And we see that our third solution here that we found for X is seven halves. So now from here, we just need to take each of these values for X and write it as an interval. So we see that the X axis can be separated into the following intervals where we first go from negative infinity to our next solution, which is negative 17 3rd. And now we know that infinity is always excluded. So our negative infinity gets closed by a parenthesis. And now we, we also know that negative 17 3rd makes our expression equal to zero. Well, since we have a strict inequality, we know that negative 17 3rd is excluded. So it gets closed by a parenthesis. And next, we go from negative 17 3rd to the next solution which is five halves. And again, both of these values for X get closed by parentheses since they are both excluded. And next, we go from five halves to our next solution, which is seven halves and both of these values are closed by parentheses. And lastly, we go from our last solution, seven halves to positive infinity and close both of these terms with parentheses. So now our next step is to assess each of these intervals and see which of them give us our proper solution. So first, we need to let F of X be equal to our left hand side of expression. So F of X is equal to the quantity of three X plus 17 times the quantity of two X minus five all divided by the quantity of two X minus seven. Cute. So now to find which interval is our solution to our inequality, we need to set up a table where we first take one of our intervals, we then choose a test value that's within the interval. And next we insert that test value, we substituted in two, we substituted into F of X which we defined above here. And lastly, we take the outcome, the, the simplified version of F of X and determine our conclusion or assess if that interval is within our solution. So we can begin with our first identified interval where we went from negative infinity to negative 3rd. And so we need to choose a test value that's within this interval. So we can choose negative six. And now we find F of negative six which is equal to the quantity of three times negative six plus 17 times the quantity of two times negative six minus five all divided by the quantity of two times negative six minus seven, all cute. And now simplifying, we see that F of negative six gives us 17 divided by the quantity of negative 19 cute. And now moving on to our solution, we see that we have a negative value. A negative outcome from our first interval. So we see that F of X, in this case is less than zero for all values of X, for all values of X within this interval of negative infinity to negative 17 3rd. So now we can move on to our next interval where we go from negative 17 3rd to halves. And now here again, we need to choose a test value here, we can just choose zero. So now we choose, we find F of zero. So substituting in zero, we have the quantity of three times zero plus times the quantity of two times zero minus five all divided by the quantity of two times zero minus seven all cute. And now simplifying, we see F of zero gives us negative 85 divided by the quantity of negative seven cute. And now we see that we will have a negative value in the numerator and the denominator which gives us a positive value. So for our conclusion, we see that F of X is going to be greater than zero for all values of X within this interval that we were testing, which goes from negative 17 3rd to positive five halves. So now we can move on to testing our next interval And our next interval will be from, from positive 5/2 to 7/2. Again, we need to choose a test value within this interval. So we can choose a whole number here. So let's say three. And now we can substitute three into F F of X. So we have F of three, which is equal to the quantity of three times three plus 17 times the quantity of two times three minus five all divided by the quantity of two times three minus seven. Cute. And now simplifying, we see that F of three is 26 divided by the quantity of negative one. Cute. So here we see that our solution is negative. So this tells us for our conclusion that F of X is going to be less than zero for all values of X within this interval where we went from five halves to seven halves. So now we are almost done here. We just have one last interval to test. So our last interval, we go from seven hands to positive infinity. And so a test value within this interval, we can just choose four. So now we find F of four and so we have the quantity of three times four plus 17 times the quantity of two times four minus five all divided by the quantity of two times four minus seven. Cute. And now simplifying, we see that F of four gives us 87 divided by the quantity of one cute. And this Is just 87 divided by one, which is just 87. So here we see that we get a positive value from our test value here. And this tells us for our conclusion that f of X will be greater than zero Will be greater than zero for all values of X within this interval where we go from 7/2 to positive infinity. So now taking a look at the four intervals that we tested, we see that for our last interval, that we have a outcome that where F X is greater than zero. And we also have a positive outcome from our second interval where we had F of X greater than zero. And we see that these two intervals sir as solutions to our original inequality because our original inequality asked us to have our expression greater than zero. So we see that the other two intervals do not work since their outcomes are negative. And so with this, we see using these two intervals, we can write our solution and we see that F of X is greater than zero for all of X, all values of X within the interval from negative 17 3rd to 5 halves And the interval. So we choose this union sign here to include the other interval where we go from 7/2 to infinity. So this is our final solution using both of these intervals. So this is our final solution set and now just scrolling back up to our answer options. We see that with these two intervals, we are left 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. (9x-11)(2x+7)/(3x-8)^3>0

Key Concepts

Rational Inequalities

Critical Points

Interval Notation

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