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

Question

Solve each rational inequality. Give the solution set in interval notation. See Examples 4 and 5. 20/(x - 1) ≥ 1

Accepted Answer

Hello, everyone. In this video, we're going to be looking at this practice problem which states for the given inequality expression solve and write the solution set in interval notation. And the inequality is 12 over X minus six is greater than or equal to six were given four answer choices for this practice problem A B C and D, all of which are intervals with differing parentheses and brackets and C and D have unions between two intervals while A and B are standalone intervals. So to determine which of these answer choices is correct, let's take a look at our inequality 12 divided by four over X minus six greater than or equal to four. So to find the solution set for this inequality, we want to solve for X and in order to solve for X, I'm going to move all my terms to the left hand side, which means I'm going to subtract by four. So when i subtract by four, I have 12 over X -6 -4 is greater than or equal to zero. Well, now I have two terms on the left hand side. And in order to combine them, I need four to have a denominator of X -6. So I'm gonna multiply four by X -6 over X -6. So this becomes -4 times X -6 over the common denominator X -6 Greater than or equal to zero. And I can simplify my numerator more by distributing this negative four. So I still have 12. Now, I have negative four, multiplied by X is negative four, X and negative four, multiplied by negative six is positive 24. And this is still over X minus six and greater than or equal to zero. I can combine like terms in my numerator, I have 12 and 24. So this becomes a negative four, X and 12 plus 24 is divided by or over X minus six Greater than or equal to zero. And from here you can see that we've simplified our expression to have all the terms on one side. But we still have X in the numerator and denominator of our expression. So in order to solve for X, we need to solve X in the numerator and the denominator. So if we solve for X in the numerator, I have negative four X plus 36 I'll set it equal to zero. I'm going to subtract 36 from both sides. So it leaves me with negative four, X is equal to negative 36 to completely isolate X. I need to divide by negative four So that leaves me with X is equal to negative 36 divided by -4, which is nine. So this is one of my solutions for X. That's only in the numerator. Now, if we look at the denominator, I have X -6 is equal to zero. Again, I'll add six to isolate X. So it leaves me with X is equal to six. So this is my second solution for X and normally these would be my answers. But because we're working with an inequality, we need to find the range of X values that satisfy this inequality. So that means that these solutions for X become the points that we need to determine possible intervals from. So I have a number line which we can call the X axis. So I have a solution at X equals six. So I'll draw a ticket for six and I'll draw another tick for nine. And based off of this interval, you can see that we have three possible. Based off of this number line, you can see that we have three possible intervals, Negative infinity to six And then 6-9 and then finally 9 to positive infinity. So in these intervals, we need to test values and see if we get inequalities that are true. So looking at the first interval, negative infinity to six, I'm going to test X equals five. So that means I'm going to plug in five from my inequality. So I have negative four multiplied by five plus 36/5 minus six is greater than or equal to zero. So if I solve this negative four multiplied by five is negative 20 plus 36 Over 5 -6, which is a negative one is greater than or equal to zero. Negative 20 plus 36 is positive 16 over negative one greater than or equal to zero. So this is saying that a negative number is greater than zero, which is not true. So I can exclude this interval from my solution set. We look at the next interval from 6 to Going to plug in the value x equals seven. So I have negative four multiplied by seven plus 36/7 minus six is greater than or equal to zero. -4 multiplied by seven is negative Plus 36/7 -6, which is one is greater than or equal to zero. Negative 28 plus 36 is equal to Positive eight still over one greater than or equal to zero. And in this case, a positive number is greater than zero, which is true. So we can include this interval in our solution set. And the final interval that we need to test is 9 to positive infinity. And I'll try x equals 10. So if we look at X equals 10, we have negative four multiplied by 10 plus Over 10 - greater than or equal to zero. Negative for multiplied by 10 is negative Plus 36/10 -6, which is four Greater than or equal to zero negative 40 plus 36 is negative 4/ greater than or equal to zero. And this is saying that a negative number is greater than zero, which is not true. So we can exclude this interval from our solution set, which means that the only interval That worked out that gave a true inequality was from 6-9. So if we go back to this number line, that would be From 6-9, and now we need to figure out if I should include The end .6 and nine. Well, because we have the greater than or equal to, I can include the end points as long as I don't get a undefined answer. And because I have a fraction, that means I can't have my denominator equaling zero. So that means that I can't include six in my solution set because when I plug in six, I get an undefined answer since my denominator will be zero. So that means that if I were to draw this on a number line, six would have a open circle And nine would have a closed circle. And if I wrote this in interval notation, six would have parentheses to nine, which would have a bracket. So this becomes my final answer for this problem. And if we look at the answer choices, you can see that answer choice. A also has a parentheses with the six going to the nine, which has a bracket. So hopefully this video helped and see you next time.

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Solve each rational inequality. Give the solution set in interval notation. See Examples 4 and 5. 20/(x - 1) ≥ 1