geno3141

If you ask for a cos, the calculator will give you a value in the range from -1 up through +1.

For example, cos(45°) = 0.7071.

If you ask for invCos, your will be entering a number in the range from -1 up through +1, and it will give you the size of the angle that has that value for a cosine.   invCos = cos-1

For example, invCos(0.7071) = 45°.

To use on a calculator, depending upon the calculator, press either the (2nd) key or the (inv) key, then cos, then enter the value.

If you change the problem to this:  a² - 10a + 21  =  0

Then, factoring:                            (a - 7)(a - 3)  =  0    --->  a = 3 or a = 7.

Going back to your original problem, an inequalty:  ≤ 0:

If you try any number (in the original problem) smaller than 3, your answer is positive; they don't work.

If you try any number from 3 through 7, your answer is negative; they all work, with the largest number = 7.

If you try any number larger than 7, your answer is positive; they don't work.

Do you mean radians per second?

If you do:  One rotation is equal to  2π  radians; so 2000 rotations is equal to  2000 x 2π  =  4000π radians.

The rate  =   4000π radians / 40 sec  =  100π radians / sec.

(z^5)^3   =   z^15

If you have an exponent to an exponent, mutiply the exponents together.

You can think of it this way, you have 3 groups and there are 5 in each group, so you have 3 x 5  =  15.

Since  z^5  =  z·z·z·z·z,

     (z^5)^3  =  (z·z·z·z·z)^3  =  (z·z·z·z·z)·(z·z·z·z·z)·(z·z·z·z·z)  =  z·z·z·z·z·z·z·z·z·z·z·z·z·z·z  =  z^15

I don't understand  -3\le x < 7

Should the  "\" be "/"?

Should  "le"  be  "ln"; if not, what is "le"?

-4c - 11  =  4c + 21

Add  4c  to both sides:

     -11  =  8c + 21

Subtract  21  from both sides:

     -32  =  8c

Divide both sides by 8:

       -4  =  c

If you have a function  f(x)  =  5, the limit will be 5.

Rewrite  8t²  ≤  3 - 10 t  as      8t² + 10t - 3 ≤ 0

Factor:                                (4t - 1)(2t + 3) ≤ 0

Solve the related equality:     (4t - 1)(2t + 3)  =  0 

     If:       4t - 1  =  0     --->     4t  =  1     --->      t  =  1/4  

     If:      2t + 3  =  0     --->     2t  =  -3     --->     t  =  -3/2 

These two points split the number line into three groups:     t ≤ - 3/2     -3/2 ≤ t < 1/4     t ≥ 1/4

Pick any number for t such that   t ≤ - 3/2;  replace it into  8t² + 10t - 3  

     and check to see if that answer ≤ 0:

     I'm going to try -10:        8(-10)² + 10(-10) - 3  =  697;  since this number isn't ≤ 0, these numbers                                                                                               don't work.

Pick any number for t such that  -3/2 ≤ t < 1/4 

     I'm going to try 0:     8(0)² + 10(0) - 3  =  -3; since this aswer is less than zero, these numbers work.

Pick any number for  t  such that     t ≥ 1/4:

     I'm goint to try 1:     8(1)² + 10(1) - 3  =  15; since this answer isn't ≤ 0, these numbers don't work.

Answer:    -3/2  ≤  t <  1/4 

Upon graphing, there is a zero at x = 6.239 (approximately).

If you use the discriminant of the quadratic formula:  b² - 4ac to determine real from complex solution:

If b² - 4ac > 0  then there will be only real.

   a = 1, b = b, and c = 15

   b² -4(1)(15) > 0

   b² > 60

   b² - 60 > 0   can be solved by factoring:

  (b - √60)(b +√60) > 0

Two cases:  

     First case: when both are positive:  

          b - √60 > 0  and  b + √60 > 0    --->     b > √60  and  b > -√60     --->   b > √60

     Second case: when both are negative:

           b - √60 < 0  and  b + √60 < 0    --->     b < √60  and  b < -√60     --->   b < -√60

These are the cases when it works; so the time that it doesn't work are:

          -√60 < b < √60

So now, list all the integers between those two radicals.