geno3141

The idea will be to get one log on each side and then take the antilog of each side.

2log(x)  =  log(x²)    <--  a multiplier goes inside as an exponent

log(2) + log(4x - 6)  =  log[2(4x-6)]   <-- logs are added when numbers are multiplied

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2log(x)  =  log(2) + log(4x - 6)

log(x²)  =  log[2(4x-6)] 

Applying the antilog to both sides:

x²  =  2(4x - 6)

x²  =  8x - 12

x² - 8x + 12  =  0

(x - 6)(x - 2)  =  0

x  =  2    or    x  =  6

Sorry, I thought there was an equal sign, not a less than sign.

If this is a democracy, I vote for CPhill's answer.

If the problem is (x² + 10x + 21) / (x +3):

Factor the numerator:  (x + 7)(x + 3)

The problem is now:   (x + 7)(x + 3) / (x + 3)

Cancel the x + 3 factor in the numerator with the x + 3 factor in the denominator, leaving  x + 7. 

Cancelling out terminating zeros in the numerator with terminating zeros in the denominator, I end with 68/20.

68/20  =  3.4

x(3x - 4)  ≤  (6x² - 3x + 5) / 10

Simplify:

3x² - 4x  ≤   (6x² - 3x + 5) / 10

30x² - 40x  ≤  6x² - 3x + 5

24x² - 37x - 5  ≤  0

Factoring:  (8x + 1)(3x - 5)  ≤  0

This divides the number line into three parts:  x < -1/8,  -1/8 < x < 5/3,  x > 5/3

Try a a number in the lower region:  I tried -1:  [8(-1) + 1][3(-1) - 5]  =  56. Since this is not less than or equal to zero, all numbers less than -1/8 will not work.

Try a number in the middle region:  I tried 0:  [8(0) + 1][3(0) - 5]  =  -5. Since this is less than zero, all numbers in this region work.    Answer:  -1/8 < x < 5/3

Try a a number in the upper region:  I tried 2:  [8(2) + 1][3(2) - 5]  =  17. Since this is not less than or equal to zero, all numbers greater than 5/3 will not work.

Now, to answer the question, you will have to figure out (count) the number of integers in the region 

-1/8 < x < 5/3.

- See more at: http://web2.0calc.com/questions/help-please_45#sthash.fDB7TCoS.dpuf

You are selecting one accountant from four, two marketing agents from three, and four board members from seven.

Since order in not important, use combinations.

4nCr1 x 3nCr2 x 7nCr4  =  4 x 3 x 35  =  420 ways

x(3x - 4)  ≤  (6x² - 3x + 5) / 10

Simplify:

3x² - 4x  ≤   (6x² - 3x + 5) / 10

30x² - 40x  ≤  6x² - 3x + 5

24x² - 37x - 5  ≤  0

Factoring:  (8x + 1)(3x - 5)  ≤  0

This divides the number line into three parts:  x < -1/8,  -1/8 < x < 5/3,  x > 5/3

Try a a number in the lower region:  I tried -1:  [8(-1) + 1][3(-1) - 5]  =  56. Since this is not less than or equal to zero, all numbers less than -1/8 will not work.

Try a number in the middle region:  I tried 0:  [8(0) + 1][3(0) - 5]  =  -5. Since this is less than zero, all numbers in this region work.    Answer:  -1/8 < x < 5/3

Try a a number in the upper region:  I tried 2:  [8(2) + 1][3(2) - 5]  =  17. Since this is not less than or equal to zero, all numbers greater than 5/3 will not work.

Now, to answer the question, you will have to figure out (count) the number of integers in the region 

-1/8 < x < 5/3.

From the first bag, you are choosing 2 out of a possible 4; that is a combination of 4 taken 2 at a time,

or  4nCr2 = 6.

Similarly, for the second bag; 

thus the answer will be  4nCr2 x 4nCr2  =  6 x 6  =  36

Although you want (n - 2)(n + 4) < 0, look at  (n - 2)(n + 4)  =  0

This is a great place to start, because it give you the two values  -4  and  2.

This divides the number line into three sections: those numbers below -4, those numbers between -4 and 2, and those numbers greater than 2.

Now, let's go back to the original inequality:  (n - 2)(n + 4) < 0.

If you try any number smaller than -4, both of the sections  n - 2  and  n + 4  will be negative, so their answer won't be < 0.

If you try any number larger than -4 and smaller than 2, one of the sections will be negative and the other will be positive, so their product will be negative, so all these numbers work.

If you try any number larger than 2, both of the sections will be positive, so their answer won't be < 0.

Answer:  -4 < n < 2.

To get the final answer, find the integers between (and not including) -4 and 2.