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Illusion [34]
3 years ago
7

-2(3x-4)=4(5x-11) Solve for x

Mathematics
2 answers:
sveta [45]3 years ago
6 0

-2(3x-4)=4(5x-11)

-6x+8=20x-44

8+44=26x

26x=52

x=52/26

your answer will be x=2

Andreyy893 years ago
3 0

Answer:

x=2

Step-by-step explanation:

-2(3x-4)=4(5x-11)

-6x+8=20x-44

8+44=26x

26x=52

x=52/26

x=2

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Prove for any positive integer n, n^3 +11n is a multiple of 6
suter [353]

There are probably other ways to approach this, but I'll focus on a proof by induction.

The base case is that n = 1. Plugging this into the expression gets us

n^3+11n = 1^3+11(1) = 1+11 = 12

which is a multiple of 6. So that takes care of the base case.

----------------------------------

Now for the inductive step, which is often a tricky thing to grasp if you're not used to it. I recommend keeping at practice to get better familiar with these types of proofs.

The idea is this: assume that k^3+11k is a multiple of 6 for some integer k > 1

Based on that assumption, we need to prove that (k+1)^3+11(k+1) is also a multiple of 6. Note how I've replaced every k with k+1. This is the next value up after k.

If we can show that the (k+1)th case works, based on the assumption, then we've effectively wrapped up the inductive proof. Think of it like a chain of dominoes. One knocks over the other to take care of every case (aka every positive integer n)

-----------------------------------

Let's do a bit of algebra to say

(k+1)^3+11(k+1)

(k^3+3k^2+3k+1) + 11(k+1)

k^3+3k^2+3k+1+11k+11

(k^3+11k) + (3k^2+3k+12)

(k^3+11k) + 3(k^2+k+4)

At this point, we have the k^3+11k as the first group while we have 3(k^2+k+4) as the second group. We already know that k^3+11k is a multiple of 6, so we don't need to worry about it. We just need to show that 3(k^2+k+4) is also a multiple of 6. This means we need to show k^2+k+4 is a multiple of 2, i.e. it's even.

------------------------------------

If k is even, then k = 2m for some integer m

That means k^2+k+4 = (2m)^2+(2m)+4 = 4m^2+2m+4 = 2(m^2+m+2)

We can see that if k is even, then k^2+k+4 is also even.

If k is odd, then k = 2m+1 and

k^2+k+4 = (2m+1)^2+(2m+1)+4 = 4m^2+4m+1+2m+1+4 = 2(2m^2+3m+3)

That shows k^2+k+4 is even when k is odd.

-------------------------------------

In short, the last section shows that k^2+k+4 is always even for any integer

That then points to 3(k^2+k+4) being a multiple of 6

Which then further points to (k^3+11k) + 3(k^2+k+4) being a multiple of 6

It's a lot of work, but we've shown that (k+1)^3+11(k+1) is a multiple of 6 based on the assumption that k^3+11k is a multiple of 6.

This concludes the inductive step and overall the proof is done by this point.

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Find each quotient. Write in simplest form. keep answer as fraction - not decimal
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So we are given the expression:

\frac{mn^{2}}{4} ÷ \frac{m^{2}n}{8}

When we divide fractions, we must flip the second term and change the sign to multiplication:

\frac{mn^{2}}{4} *\frac{8}{m^{2}n}

And then we multiply across:

\frac{mn^{2}}{4} *\frac{8}{m^{2}n}= \frac{8mn^{2}}{4m^{2}n}

Then we can break apart all of the like variables for simplification:

\frac{8mn^{2}}{4m^{2}n}=\frac{8}{4} * \frac{m}{m^{2}} *\frac{n^{2}}{n}

When we simplify variables through division, we subtract the exponent of the numerator from the exponent of the denominator. So we then have:

\frac{8}{4} =2

\frac{m}{m^{2}}=m^{-1} =\frac{1}{m}

\frac{n^{2}}{n}=n^{1}=\frac{n}{1}

So then we multiply all of these simplified parts together:

\frac{2}{1}*\frac{1}{m}*\frac{n}{1}   =\frac{2n}{m}

So now we know that the simplified form of the initial expression is: \frac{2n}{m}.

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Use the Algebra Tiles to construct the algebraic expression for this situation:
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That’s a hard one use Brainly to find the answer
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