Use the method described in Exercises 83–86, if applicable, and p...

Question

Use the method described in Exercises 83–86, if applicable, and properties ofabsolute value to solve each equation or inequality. (Hint: Exercises 99 and 100 canbe solved by inspection.) | 3x^2 + x | = 14

Accepted Answer

Hey everyone here we are asked to use absolute value properties. To solve the following inequality where we have the absolute value of four X squared plus three X is equal to 22. Now our first step here in solving this problem is to recall the absolute value property where if the absolute value of X is equal to a. Where A. Is greater than zero, then we know that X is equal to A. Or X is equal to negative A. So using this property we can take the expression within the absolute value signs which is four X squared plus three X. And set it equal to positive 22 negative 22. So beginning with positive 22 again we take four X squared plus three X. And now set it equal to positive 22. Well now we just need to solve for X. So we first just need to subtract 22 from both sides. So we have four X squared plus three X minus 22 is equal to zero. Now to solve for X, we can just factor this expression on the left hand side. So we have the quantity of four X plus 11 times the quantity of x minus two is equal to zero. Now we can find our values for X by setting each set of parentheses equal to zero. So we can start with the first part of our expression where we have four X plus 11. And again setting it equal to zero. We need to solve for X. So we have we first need to subtract 11 from both sides. So we have four X is equal to negative 11. Now we need to divide both sides by four to get x comps completely alone. And we see that X is equal to negative 11 4th. Now we can move on to the second part of our expression where we have x minus two and again we need to set it equal to zero. Now solving for X, we need to add two to both sides and we see X is also equal to positive two. So now we have found two solutions for X by setting our expression equal to positive 20 two. So now we need to take the same expression and set it equal to negative 22. So again we have four X squared plus three, X is equal to now negative 22. So now we again need to solve for X. So first we need to add 22 to both sides and we have four X squared plus three X plus is equal to zero. Now to find our values for X. For this expression we can use the quadratic formula. So if we recall the quadratic formula, we have X is equal to negative B plus or minus the square root of b squared minus four. A. C. All over To a. So now with our expression we see that our value for a is going to be for our value for B is going to be three and our value for C is just 22. So now we can just plug in these values into our quadratic formula. So we have X is equal to negative three plus or minus the square root of three squared minus four times four times 22. All over two times 4. Now we just need to simplify. So we have X is equal to negative three plus or minus. The square root of nine minus 352 all divided by eight. And now continuing to simplify, we see we have X is equal to negative three plus or minus the square root of nine minus 352 gives us negative 343 and this is all divided by eight. And now again we need to continue to simplify our route here. So we have X is equal to negative three plus or minus Now the square root of simplifies to seven times the square root of seven, but the square root of negative one is I. So we have seven times the square root of seven times I And this is all divided by eight. And so now we have simplified this expression as much as we can. So we see that we have found two more solutions for X. We have negative three plus or minus seven times the square root of seven times I. All divided by eight. So now we can write out all of our solutions for X as a solution set. So we have the cell solutions for X as negative 11 divided by four. We also found X to be equal to two. And lastly we found X to be equal to negative three plus or minus seven times the square root of seven times I all divided by eight. And with this we have found four solutions four X. And so we are left with answer choice D. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

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Use the method described in Exercises 83–86, if applicable, and properties ofabsolute value to solve each equation or inequality. (Hint: Exercises 99 and 100 canbe solved by inspection.) | 3x^2 + x | = 14

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