How do I solve an inequality for x with this problem −12x−2x2>−14−12x−2x2>−14?

Answer 1

Putting into standard order

-14 < -2x² - 12x - 14 < -2x² - 12x

12x - 14 < -2x² - 14 < - 2x² add 12x to all

2x² + 12x - 14 < -14 < 0 add 2x² to all

2x² + 12x - 14 < -14 If < -14 then < 0

2x² + 12x < 0 add 14 to both sides

x² + 6x < 0 divide both sides by 2

Can you complete and answer?

Answer 2

Hey Kailey, I think you're asking this question:

Solve for x:

-12x - 2x² > -14 - 12x - 2x² > -14

If that's correct, make sure when typing on a conventional keyboard, you type the question as

-12x-2x^2>-14-12x-2x^2>-14

that way, the reader won't thinking you're multiplying two and two with "2x2".

Ookay, to the problem now

First, remember that this is an inequality, which is similar to an equation in that you are allowed to:

1) add/subtract a number from each side (here we have 3 sides) of the equation/inequality

2) multiply or divide each side by a number (provided that number is not zero and that no variables cancel)

3) raise each side to a power, especially to the one half, or (1/2) (or .5 depending on your comfort level with fractions)

But inequalities are different from equations in that when we multiply or divide by a negative number, we have to switch the orientation of the inequality.

So with that said, let's use those rules on the problem

-12x -2x^2 > -14 -12x - 2x^2 > -14

We have a three sided inequality here, and I see we have some constant terms (the -14's), some x terms (the -12x's, and even some x^2 terms. This means we will probably be solving a quadratic soon. When we know we'll have to solve a quadratic, the first goal is to add or subtract all of the terms to one side of the equation so that the other side is zero. But here, we have a three sided inequality, so how would we possibly get all terms on one side, there are three sides! Fortunately, in this question we get lucky: it will be possible to move all x and x^2 terms to one side of the inequality:

TL:DR: because we see x^2 terms, need to solve quadratic by getting all terms to one side using addition and subtraction, then factor and either consider the results graphically on an x y plane or a number line

Right, all of the terms to one side, First, let's move the 2x² terms to one side. Add 2x^2 to all sides.

-12x -2x² > -14 -12x - 2x² > -14

+2x² + 2x² +2x²

Where bold text is text that you would write down underneath the question statement as a way of keeping track of your work.

-12x > -14 -12x > -14 + 2x²

Now, let's add 12x to all sides

- 12x > - 14 - 12x > -14 + 2x²

+ 12x + 12x + 12x

to get

0 > -14 > -14 + 2x² + 12x

That was pretty convenient. Almost all of the terms are now all on the right side. In fact, we actually have a quadratic on the right side already. This three way inequality is saying that 0 is bigger than -14 (thanks, got it) which is bigger than our quadratic. At this point, we can just write

-14 > -14 + 2x² + 12x

and now we can totally get all of the terms to one side, lets add 14 to each side

-14 > -14 + 2x² + 12x

+14 +14

to get

0 > + 2x² + 12x

and now we can factor the right hand side.

0 > 2x (x + 6)

and see that if this were an equation (which it isn't, but if it were) then the solutions would be x = 0 and x = -6) But, since we have an inequality, you could graph y = 2x(x+6) and see that it is a quadratic that "opens up" (has a downward pointing vertex) that crosses through the origin and the point x = -6, meaning that it is above the x axis when x < -6 and x > 0 and beneath the x axis on - 6 < x < 0. So, because our quadratic is less than zero in this question, the correct interval is - 6 < x < 0.

I hope that helped Kailey.