Quantum equation predicts universe has no beginning February 9, by Lisa Zyga, Phys. Note on the left the dramatic expansion not to scale occurring in the inflationary epoch, and at the center the expansion acceleration. The scheme is decorated with WMAP images on the left and with the representation of stars at the appropriate level of development. The model may also account for dark matter and dark energy, resolving multiple problems at once.

Absolute value equations Video transcript So we're asked to solve for x, and we have this equation with absolute values in it. So it's 4 times the absolute value of x plus 10 plus 4 is equal to 6 times the absolute value of x plus 10 plus And at first, this looks really daunting, but the key is to just solve for this absolute value expression and then go from there.

Let me just rewrite it so that the absolute value expression really jumps out.

So this is 4 times the absolute value of x plus 10 plus 4 is equal to 6 times the absolute value of x plus 10 plus So let's get all of the absolute values of x plus 10 on the left-hand side. So I want to get rid of the 6 times the absolute value of x plus 10 on the right. Well, how would I do that?

Well, I could subtract 6 times the absolute value of x plus 10 from the right, but we've already seen this multiple times. If these two things are equal, and if I want to keep them equal, if I subtract 6 from the right-hand side, I've got to subtract-- or if I subtract 6 times the absolute value of x plus 10 from the right-hand side, I have to subtract the same thing from the left-hand side.

So we're going to have minus 6 times the absolute value of x plus And likewise, I want to get all my constant terms, I want to get this 4 out of the left-hand side.

So let me subtract 4 from the left, and then I have to also do it on the right, otherwise my equality wouldn't hold. And now let's see what we end up with. So on the left-hand side, the 4 minus 4, that's 0. You have 4 of something minus 6 of something, that means you're going to end up with negative 2 of that something.

Negative 2 of the absolute value of x plus Remember, this might seem a little confusing, but remember, if you had 4 apples and you subtract 6 apples, you now have negative 2 apples, I guess you owe someone the apples.

Same way, you have 4 of this expression, you take away 6 of this expression, you now have negative 2 of this expression. Let me write it a little bit neater. So it's negative 2 times the absolute value of x plus 10 is equal to, well the whole point of this, of the 6 times the absolute value of x plus 10 minus 6 times the absolute value of x plus 10 is to make those cancel out, and then you have 10 minus 4, which is equal to 6.

Now, we want to solve for the absolute value of x plus So let's get rid of this negative 2, and we can do that by dividing both sides by negative 2. You might realize, everything we've done so far is just treating this red expression as almost just like a variable, and we're going to solve for that red expression and then take it from there.

So negative 2 divided by negative 2 is 1. So we get the absolute value of x plus 10 is equal to negative 3. Now, this gets us to a very interesting situation.

You might say maybe this could be the positive version or the negative, but remember, absolute value is always non-negative. If you took the absolute value of 0, you would get 0.Readbag users suggest that Solutions to Time Series Analysis With Applications in R, second edition is worth reading.

The file contains page(s) and is free to view, download or print. kcc1 Count to by ones and by tens. kcc2 Count forward beginning from a given number within the known sequence (instead of having to begin at 1). kcc3 Write numbers from 0 to Represent a number of objects with a written numeral (with 0 representing a count of no objects).

kcc4a When counting objects, say the number names in the standard order, pairing each object with one and only. kcc1 Count to by ones and by tens. kcc2 Count forward beginning from a given number within the known sequence (instead of having to begin at 1).

kcc3 Write numbers from 0 to Represent a number of objects with a written numeral (with 0 representing a count of no objects). kcc4a When counting objects, say the number names in the standard order, pairing each object with one and only.

In this section we solve linear first order differential equations, i.e. differential equations in the form y' + p(t) y = g(t). We give an in depth overview of the process used to solve this type of differential equation as well as a derivation of the formula needed for the integrating factor used in the solution process.

This is an artist's concept of the metric expansion of space, where space (including hypothetical non-observable portions of the universe) is represented at each time by the circular sections. Follow these steps to solve an absolute value equality which contains one absolute value: Isolate the absolute value on one side of the equation.

Is the number on the other side of the equation negative? If you answered yes, then the equation has no solution. If you answered no, then go on to step 3. Write two equations without absolute values.

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No Big Bang? Quantum equation predicts universe has no beginning