Solve each inequality. Give the solution set using interval no...

Question

Solve each inequality. Give the solution set using interval notation. $\frac{3 x - 2}{x} - 4 > 0$

Accepted Answer

Hey, everyone here, we are asked to find the solution set of the following inequality and express it in interval notation. And we are given the inequality where we have the quantity of four X minus seven divided by X minus 11 is greater than zero. So now our first step here in solving this inequality is to first combine our whole number with our fraction here. So we need to get a common denominator. And so all we have to do is multiply our whole number by the value in the denominator of the fraction. So we need to multiply the numerator and the denominator of by X. So 11 is multiplied by X divided by. And now simplifying we have the quantity of four X minus seven divided by X minus 11 X divided by X is all greater than zero. So now that we have a common denominator, we can combine our numerator hers and we have four X minus seven minus 11 X all divided by X is greater than zero. So now we just want to combine our like terms to simplify our expression here. So doing this for X minus 11 X gives us negative seven X minus seven all divided by X is greater than zero. So now we want to solve this inequality by setting the numerator and the denominator individually equal to zero. So we can start with the numerator where we have negative seven X minus seven. And again, we set this expression equal to zero. And now we want to solve for X. So first, we just need to add seven to both sides and we get negative seven, X is equal to positive seven. And now we need to get rid of our coefficient. So we just divide both sides by -7. And we see that one of our values or solutions for X is just negative one. So now we can repeat this with the expression in the denominator which is just X. So while we take the denominator and we set it equal to zero, so we see that the second solution for X is just zero. So now these solutions that we obtained can tell us that the X axis can be separated into multiple intervals. So we first want to go from negative infinity to our smallest solution here, which is negative one. And both of these terms are going to be closed by parentheses since we know that infinity is always excluded and negative one causes the expression to be equal 20. Well, since we have a strict inequality, we know that we cannot include this solution. And then we also can create another interval where we go from the solution that we had, which was negative one to the next solution, which is zero. So both of these terms again are closed by parentheses. And then we go from our last solution to positive infinity. And once again, both of these terms are closed by parentheses. So now we see that the X axis can be separated into these intervals that we found. So we now need to test values in each of these intervals to see which interval gives us the correct solution set for our given inequality. So we want to first let F of X B be equal to the left side of our inequality where we had the quantity of negative seven X minus seven all divided by X. So now again, we want to start testing our, our intervals here. So we're going to set up a table where first we take one of our intervals that we identified and then we input a test value that's within the interval. And we take this test value and place it or substitute it into our function here. So we substitute it in two F of X. And then lastly, this result from FF X allows us to make some conclusion with our interval to see if this interval is a solution to our inequality or not. So we can start with our first identified interval where we went from negative infinity to negative one And now we can choose any test value within this interval. So we can go ahead and choose just let's say negative two. So now we need to find F of negative two and F of negative two gives us negative seven times negative two minus seven identified by negative two. And simplifying we get negative seven divided by two as our answer here. So we see that we have a negative solution which allows us to form a conclusion where we know that F of X for this interval is going to be less than, than zero for all X terms for all X terms in the, the interval that we tested, which is from negative infinity to negative one. So now we need to repeat this process for the rest of our intervals. So we can move on to our second identified interval where we went from negative 1 to 0. So again, we need to choose a test value here. We can choose negative one half. And now we find F of negative one half which we find is negative seven times negative one half minus seven, all divided by negative one half. And now simplifying we get just positive seven. So since we obtained a positive value, we can form a conclusion where we know that F of X is going to be greater than zero for all X values that are with in the interval which we tested, which is from negative 1 to 0. So now we just want to test our last interval where we went from zero to positive infinity. Again, we need to choose a test value here, we can choose just positive one. So now we find F of positive one and now substituting in one into F of X, we find negative seven times one minus seven all divided by one. And now simplifying we get the solution of negative 14. So now with this negative value, we know that F of X is going to be less than zero for all X values that are within the interval that we tested, which is from zero to positive infinity. So now that we have tested each of our intervals, we again want to recall our original inequality where we had the function, we had F of X and we needed it to be greater than zero. So now we need to look at our conclusions and see which of our intervals give us F of X is greater than zero. And here looking, we see the first interval gives us a negative value. So we know this interval is not a solution. But our second interval does give us a positive value. FFX is greater than zero. So this is a solution. And again, our last interval gives us a negative value. So we know that This is not a solution. So with this, we find that our solution set to our given inequality is the interval from negative 1-0 and both of these terms are closed by parentheses. So with this, we have found our solution set with which leaves us with answer choice. D thanks for watching. I hope you found this video helpful and I'll see you again next time.

Solve each inequality. Give the solution set using interval notation. $\frac{3 x - 2}{x} - 4 > 0$

Key Concepts

Rational Inequalities

Critical Points and Sign Analysis

Interval Notation

Watch next

Solve each inequality. Give the solution set using interval notation. 3x−2x−4>0\frac{3x - 2}{x} - 4 > 0

Key Concepts

Rational Inequalities

Critical Points and Sign Analysis

Interval Notation

Watch next

Solve each inequality. Give the solution set using interval notation. $\frac{3 x - 2}{x} - 4 > 0$