geno3141

2√x  >  2⁷

2√x÷2  >  2⁷÷2

√x  > 2⁶

(√x)²  >  (2⁶)² 

x  >  2¹²

4x² + 8x + 3  =  0

First, move the number term to the other side:

Subtract 3 from both sides                    4x² + 8x      =  -3 

Then, divide both sides by the coefficient of the squared term:

Divide all the terms by 4:                        x² + 2x       =  -3/4

Now, three steps:  divide the coefficient of the linear terms by 2:     2/2  = 1

                                                                    square this result:           1²  =  1

                              add this number to both sides of the equation:

                                                                   x² + 2x + 1  =  -3/4 + 1

Factor the left-hand side; simplify the right side:

                                                                           (x + 1)²  =  1/4

Find the square root of both sides:                        x + 1  =  ± 1/2

Subtract 1 from both sides:                                         x  =  -1 ± 1/2     --->      x  =  -1.5  or  x  =  0.5

For the second question, the interest rate of 6% yields $24,800.00 while the interest rate of 7% yields $26,398.59.  The question asks how many more dollars the larger yield is than the smaller yield (subtract!).

For the first question:  12²  =  (4·3)²  =  (2·2·3)²  =  (2²·3)²  =  2⁴·3²

                                    18³  =  (2·9)³  =  (2·3·3)³  =  (2·3²)³  =  2³·3⁶

12²·18³  =  (2⁴·3²)·(2³·3⁶)  =  2⁷·3⁸

Since   12²·18³  = 2^x·3^y     --->    2⁷·3⁸  =  2^x·3^y     --->     x = 7   and   y = 8 

There are a lot of steps here; any question?

|x² - 14x + 29| = 4

Either:  x² - 14x + 29  =  4         or         x² - 14x + 29  =  -4

             x² - 14x + 25  =  0         or         x² - 14x + 33  =  0

You can factor the one on the right but use the quadratic formula on the left one.

There will be four solutions, two from each part.

Either     |2 - c|  =  +(2 - c)  =  2 - c     or     |2 - c|  =  -(2 - c)  =  -2 + c

So:  |2 - c| + 2c  =  13     --->     |2 - c|  =  13 - 2c

If  |2 - c| = 2 - c               --->      2 - c    =  13 - 2c   --->   2 + c  =  13   --->   c  = 11

If  |2 - c| = -2 + c            --->     -2 + c   =  13 - 2c   ---?   -2 + 3c = 13   --->   3c  =  15   --->   c  =  5

For any number,  n,  |n|  is positive.

So, if  n  is positive, then  n/|n|  is  +1  and if  n  is negative, then  n/|n|  is  -1.

For xyz/|xyz|:

--  if all three, x, y, and z, are negative, then  xyz  is negative, and  xyz/|xyz|  =  -1.

-- if two of them are negative, then  xyz  is positive, and  xyz/|xyz|  =  +1.

-- if one of them is negative, then  xyz  is negative, and  xyz/|xyz|  =  -1.

-- if none of them is negative, then  xyz  is positive, and  xyz/|xyz|  =  +1.

If x is negative, y is negative, and z is negative,

   then x/|x| + y/|y| + z/|z| + xyz/|xyz|  =  -1 + -1 + -1 + -1  =  -4

If x is positive, y is negative, and z is negative,

   then x/|x| + y/|y| + z/|z| + xyz/|xyz|  =  +1 + -1 + -1 + +1  =  0

If x is negative, y is positive, and z is negative,

   then x/|x| + y/|y| + z/|z| + xyz/|xyz|  =  -1 + +1 + -1 + +1  =  0

If x is negative, y is negative, and z is positive,

   then x/|x| + y/|y| + z/|z| + xyz/|xyz|  =  -1 + -1 + +1 + +1  =  -2

If x is positive, y is positive, and z is negative,

   then x/|x| + y/|y| + z/|z| + xyz/|xyz|  =  +1 + +1 + -1 + -1  =  0

If x is positive, y is negative, and z is positive,

   then x/|x| + y/|y| + z/|z| + xyz/|xyz|  =  +1 + -1 + +1 + -1  =  0

If x is negative, y is positive, and z is positive,

   then x/|x| + y/|y| + z/|z| + xyz/|xyz|  =  -1 + +1 + +1 + -1  =  0

If x is positive, y is positive, and z is positive,

   then x/|x| + y/|y| + z/|z| + xyz/|xyz|  =  +1 + +1 + +1 + +1  =  4

It is easier for us if you gave us specific questions ...

Perhaps ... what is the question?

(3n + 3)!  =  (3n + 3)·(3n + 2)·(3n+1)·(3n)!

Write into exponential notation:  base = 3,   exponent = 2,  answer = log₂x

log₃(log₂x) = 2   --->   3²  =  log₂x   --->   log₂x  =  9

Now, write this into exponential notation:  base = 2,   exponent = 9,  answer = x

log₂x  =  27   --->  2⁹  =  x   --->  x = 512