Solve each polynomial inequality in Exercises 1–42 and graph the ...

Question

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. 9x^2−6x+1<0

Accepted Answer

Hello. Today we're gonna be factoring the given polynomial inequality. So what we are given is 36 X squared minus 132 X plus 121. Less than zero. Now, in order to solve for this inequality, we're going to need to completely factor the left hand side of the inequality first here. What we are given is a fact herbal. Try no meal. So in order to factor this, Trin Amiel, we're going to need to use the trial and error method and we're going to start that by listing the factors of the first term and the last term, The factors of 36 are going to be one in 36 six times six and negative six times negative six. And the factors of the last term are going to be one in 11 times 11 and -11 times negative 11. Now, when we factor this, the result is going to be the product of two groups. The first factors are going to be called A factors and the second factors are gonna be called the B factors. So when we put this into the two groups, this is going to be a X plus or minus B times a X plus or minus B. And this is going to be where the A factors are placed first and the B factors come afterward. So what's the pair of factors that we can choose to multiply together? That will result in the original Tryin Amiel. Well let's go ahead and pick six and six and negative 11 and negative 11. If we plug this into this form, the factored form is going to be six x minus 11 times six X minus 11 and we're gonna go ahead and check to make sure that this foils to the original Tryin Amiel. So again we chose six x minus 11 times six x minus 11. If we follow this together we're gonna do six times six which gives us 36 X squared six times negative 11 which gives us negative 66 X negative 11 times six X which is negative 66 X and negative 11 times negative 11 which is going to give us positive 121 combining like terms is going to give us X squared minus 132 X plus 121 which matches the original Try no meal. So six x minus 11 times 66 minus 11 is going to be the factored form of the original Try no meal and this is going to be less than zero. So here we have six x minus 11 but this factor pops up twice. We can combine this into a single factor to give us six x minus squared and that's going to be less than zero. Now that we've completely factored the inequality, we're going to need to plot these values on the number line to see what regions we can include as solutions to the inequality but in order to do that, we're going to need to first find the zero of X in order to find the zero of X. We're going to take the factors of X and set it equal to zero and solve. So doing that is going to give me six, X minus 11 squared is equal to zero. In order to solve for X. I'm going to take the square root of both sides of this equation and this is going to give me six. X minus 11 is equal to the square root of zero, which is zero. I'm going to add 11 to both sides and that's going to give me six. X is equal to 11. And finally, if I divide both sides by six, this is going to give me X is equal to 11/6 and this is going to be the zero for X. Now let's go ahead and plot this on a number line. So we're gonna draw a number line which goes from negative infinity to our first zero, which is 11/6 and from 11/6 to positive infinity and notice that the inequality uses the less than sign, since it's using the left hand side, we don't want to include the zero in our set of our set of solutions. So we're going to represent it with an open circle. So now what we want to go ahead and do is test the regions around 11/6. Now, one thing to note, 11/6 is approximately equal to 1.84. So it lies between one and two on the number line. So we can go ahead and use one and two as testing points to figure out what region satisfies the factored form of the inequality. If we let x equal to one and plug it into our factored form of the inequality, what this gives us is six times 1 -11 squared and that should be equal to zero. Six times one is going to give us six and six minus is going to give us negative five. And finally negative five squared is going to give us positive 25 that's going to be less than zero. But this is not true because any positive number cannot be less than zero. So this inequality is false. Thus, we are not including any values that come before 11/6. Next we're going to pick X is equal to two as a testing point. And if we plug this into our factored inequality, we get six times two minus 11 squared less than zero. Six times two gives us 12 and 12 minus 11 gives us one. So we have one squared less than zero and one squared is going to give us one which is less than zero, but one cannot be less than zero. So that's going to be false. So that means that we're not gonna include any values to the right of 11/6 on the number line. So in order to interpret this, none of the values before 11/6 satisfy the inequality, and none of the values that come after 11/6 satisfy the inequality. And we are not including the value 11/6 in our solutions to the inequality. Thus no number that we plug in for X is going to make the inequality true, which means that there's no solution to the given inequality. And that means that the answer to this problem is going to be D. So I hope this video helps in understanding how to approach this problem, and I'll go ahead and see you all in the next video.

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Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. 9x^2−6x+1<0