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

Question

Solve each inequality. Give the solution set using interval notation. 3x+6 / x-5 > 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 quality of X plus 30 all divided by the quantity of X minus seven. And this is greater than zero. So now to find the solution set of this inequality, we first want to find values of X which caused our expression on the left side to be equal to zero. So that means we just want to set the numerator and the denominator individually equal to zero to obtain our X values. So we can start with our numerator where we have 15 X plus 30. And again, we want to set this expression equal to zero. So now we need to solve for X. So first we subtract 30 from both sides to get our X term alone. And this gives us 15 X is equal to negative 30. And now we need to get rid of the coefficient by dividing both sides by 15. And we see that one of our solutions for X is negative two. So now we can repeat this with the denominator of our original expression, which was X minus seven. And again, we set this expo equal to zero. Now solving for X, all we need to do is add seven to both sides. And we see that the other solution for X is positive seven. So now with these solutions for X, we can rewrite them as intervals and it tells us that the X axis can be separated into the intervals where we go first from negative infinity to the smallest solution which is negative two. And for now we close both of these terms by parentheses. And then we go from negative two to the next solution which is seven, which both of these terms are closed by parentheses. And then we go from seven to positive infinity. And once again, both of these terms are closed by parentheses. So with these intervals, we now need to test each of the intervals to see which of them satisfy our given inequality. So first, we need to let the function F of X be equal to the left side of our inequality, which is the quantity of 15 X plus 30 divided by the quantity of X minus seven. So now we can use this function to test our intervals. So we first just want to set up a table where first we take one of our intervals and then we need to choose a test value, a test value that is within the chosen interval. And then with this test value, we substitute it in to the function F of X. And then we obtain a value of which we can assess to form a conclusion of the interval based on the sign of the value. So we can begin by choosing our first identified interval where we go from negative infinity to negative two. And now again, we need to choose a test value within this interval. So we can choose negative three. And now we need to find F of negative three well inputting or substituting in negative three into our function, we get 15 times negative three plus 30 all divided by negative three minus seven. And now simplifying, we see that F of negative three gives us positive three halves. Well, from our positive value that we obtained from our test value, we can conclude that F of X is going to be greater than zero for all X values within the interval where we go from negative infinity to negative two. So now we have finished testing our first interval. So we can move on to our, our next interval where we go from negative to, to positive seven. And now a test value within this interval can be zero. So now we need to find F of zero. And so we have 15 times zero plus 30 all divided by zero minus seven. And now simplifying, we see that F of zero gives us negative 30 divided by seven. So with this negative value, we can conclude that F of X is going to be less than zero for all X values within the interval, negative 2 to 7. So now we can test our last interval where we go from seven to positive infinity. And so now we choose a test value within this interval and we can choose positive eight. So now we need to find F of positive eight. And so we see that we have 15 times eight plus 30 all divided by eight minus seven. And now simplifying we get 150 divided by one which is just 150. And from this positive value, we can conclude that F of X is going to be greater than zero for all X values within the interval from seven to positive infinity. So now from each of our conclusions, we can see which of our intervals satisfy our original inequality where we were trying to find where X is greater than zero. So looking at our conclusions, we see that F of X is greater than zero from negative infinity to negative two as well as from seven to positive infinity. So we can see that X is going to be greater than zero for all X values from negative infinity to negative two and from seven to positive infinity. So with this, we have found the solution set for our given inequality. And so 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 inequality. Give the solution set using interval notation. 3x+6 / x-5 > 0

Key Concepts

Solving Rational Inequalities

Critical Points and Sign Analysis

Interval Notation

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