Use the technique described in Exercises 87–90 to solve each ineq...

Question

Use the technique described in Exercises 87–90 to solve each inequality. Write the solution set in interval notation. 2x^2 - 9x ≥ 18

Accepted Answer

Hello, everyone. In this video, we're going to be looking at this practice problem which states find the solution for the given quadratic inequality expressed the answer in interval notation. And the quadratic inequality is four x To the power of 2 -29 x greater than or equal to 63. And were given four answer choices, all of which are intervals and all the intervals seem to have this union and nine and 7 4th seemed to be popular numbers here, whether they have a negative in front of them or not. And you can see that within these answer choices, we can see that the inner components always have brackets and the outside parentheses. And in order to figure out which of these answers choices is good, let's take a look at our quadratic inequality. So for quadratic inequality, we have four X squared minus 29 X greater than equal to 63. So in this case, you can see that we have almost a quadratic equation. So in order to try to solve for X in, I'm gonna move all the terms over to the left hand side. So I'm going to subtract 63 from both sides. So I'm left with four X squared -29 X -63 is greater than or equal to zero. So now that I have a, all my terms on one side, I'm going to try to sulfur X by factoring. So when I factor I'm going to have two terms and I need the first term of my factors to multiply 24 X squared. So when I think of factors of four X squared, I can do four X times X, two, X times two X. And for these factors, I'm going to try four x times x and then I want my second terms To multiply to give me -63. So I think of the factors for 63 or in this case -63, I can have positive seven and -9 or positive nine and -7. So in this case, I'm going to try for X -7 plus and then my second one will be X plus nine. So now with these factors, I can make sure that they're correct by multiplying them again and making sure I get back my original equation. In this case, four X times X is four X squared, four X multiplied by nine is X negative seven, multiplied by X is negative seven, X, negative seven multiplied by positive nine is negative 63. So I combine my middle terms, I have 36 X minus seven X, so this becomes positive 29 X. So in this case, you can see that I have almost the same exact equation except that my middle term is positive instead of negative. So that means I'm going to switch the signs for my second term. So instead of being negative seven, I'm going to do positive seven and a negative nine. So now when I do this multiplication, I have four X multiplied by X still gives me four X squared four, x multiplied by -9 is now -36, X Positive seven multiplied by X is seven X And then seven multiplied by -9 is still -63. So now you can see that when I combine my middle terms, I'm left with negative 29 X which is my original equation. So I know that this factoring this factory ization is good. And from here, I'm going to want to solve for X in each of these factors. So I'm gonna say four, X plus seven is equal to zero And X -9 is equal to zero. So have I subtract seven in this first factor with four, X is equal to negative seven To isolate X. I'm going to divide by four. So X is equal to negative 7/ solving for my second factor. I'm going to add nine to both sides and I get X is equal to nine. So you may be thinking these are the final answers. For X. But recall that we are looking at an inequality. So we're looking for the range of X values that will make this inequality true. So what these points are telling us X equals negative 7/4 and X equals nine is the points that we need to be looking at the intervals we need to be looking at. So if I were to draw a number line R G X axis, I'll draw a tick at negative 7/4 and another tick at positive nine. And these are the intervals that we need to be looking at. So in each of these intervals, I need to see if The inequality is true. So this first interval is negative infinity to negative 7/4. The second interval is negative 7/4 to positive nine. And my last interval is from nine to positive infinity. So if I test a point at this first interval from negative infinity to negative 7/4, say I test x equals negative three. So that means I'm going to plug in negative three into my equation. So I have four, negative three squared minus Multiplied by -3 minus 63 is greater than or equal to zero. So negative three squared is nine. So now how four times 9 is 36? Now I have negative 29 multiplied by -3. That's two positives make two negatives make a positive and 29 multiplied by three is Stalled the -63 greater than or equal to zero. If I add 36 plus 87, That is equal to 123. So now I have 123 -63, that is equal to 62, saying 60 is greater than or equal to zero. And this is a true statement. So I know that I can include this interval in my solution set. Looking at our second interval from negative 7/4 to positive nine, I'm going to try X equals zero. So when I plug in zero for X, I have four times zero squared -29, multiplied by zero -63 is greater than or equal 20, four times zero squared to 0. 29 multiplied by zero is zero with negative 63 is great event or equal to zero. And this is not true. So I don't need to include this interval in my solution set. So the final interval we're looking at is nine to positive infinity and I'm going to test x equals 10. So if I try x equals, 10 I'm going to have four multiplied by squared -29, multiplied by 10 -63 is greater than or equal to zero. 10 squared is 100. So four times 100 or 400 -29 multiplied by 10, which is 290 -63, 400 -2 90 is 110. And now I have 110 -63, which is equal to 47. So 47 is greater than, or equal to zero. And once again, this inequality is true. So I want to include this interval in my solution. So the only interval that didn't make it is the center here. So if I were to draw the intervals that did make it on this number line, I have X going from negative 7/4 to negative infinity and X from nine To positive infinity. But we need to figure out if I need to include negative 7/4 and nine. So I'm gonna plug in -7/4 and see if it makes the inequality true. So I have four times negative 7/4 Squared -29 multiplied by negative 7/ minus 63 -7/ squared is equal to 49 over 16. So now four times 49/16 in the council of fours out. So I'm left with 49/4. Now I have negative 29 multiplied by negative 7/4. 2 negatives make a positive and 29 multiplied by seven is 203. And we still have the denominator of four. And I'm going to turn this 63 to have a denominator of four. So if I do that 63 is 252 divided by four. So these all have the same denominator which means that I can combine them. And when I combine them, I get 0/4 and recall that inequality is greater than or equal to zero. And in this case is saying zero is greater than or equal to zero, which is a true statement. So I can include negative 7/4 in my solution. So I go back to my number line that would show up as a closed circle. Now we can test x at nine. So x equals nine, I have four times nine squared minus 29 times nine minus 63 greater than or equal to squared is nine times nine, which is 81 Or multiplied by 81 is 324. Then we have 29 multiplied by nine, Which is 261 - Is greater than or equal to zero. And once I combine all of these numbers, I get zero is greater than or equal to zero, which is a true inequality. So I can also include X equals nine in my solution set. So if I went back to this number line, I would also have a closed circle here. And if I wrote that in interval notation, X goes from negative infinity to negative 7/4 and 7/4 has a bracket because X can equal seven, negative 7/4. And that's union where the second interval from nine, which another bracket because X can equal nine to positive infinity. So this becomes my final answer for the solution. And if we look at the answer choices, you can see that answer choice C also has the same interval. So hopefully this video helped and see you next time.

Use the technique described in Exercises 87–90 to solve each inequality. Write the solution set in interval notation. 2x^2 - 9x ≥ 18

Key Concepts

Inequalities

Quadratic Functions

Interval Notation

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