geno3141

This is a combinations problem.

The number of ways to choose 3 of the first 5 problems is:  ₅C₃ = 10.

There are 9 problems remaining (14 - 5 = 9) and six more problems must be chosen (9 - 3 = 6); the number of ways to choose 6 of these 9 problems is:  ₉C₆ = 84.

Since both of these conditions must be met, multiply these numbers together:  10 x 84  =  840.

The absolute value function guarantees that the value must always be zero or positive.

x can be any value: positive, zero, or negative but y can never be any negative number.

Examples:

---   if x = 5, then y = |x| = |5| = 5

---   if x = 0, then y = |x| = |0| = 0

---   if x = -5, then y = |x| = |-5| = 5

The hare is walking 4 feet/min faster than the tortoise.

To gain 100 feet, there will have to be:  100 ft ÷ 4 ft/min  =  25 minutes.

3x² + x²

These are "like terms" -- they have exactly the same variables, so add their coefficients:  3 + 1 = 4

Answer:  4x²

THe legnths of any two sides of a triangle, when added together, must be longer than the third side.

So:  10 + x > 12   --->   x > 2   --->   2 < x

And:  10 + 12 > x   --->   22 > x   --->   x < 22

And:  12 + x > 10   --->   x > -2   (Since, by the first inequality, x > 2; for x to be both > -2 and > 2, it must                                                               be > 2, so we'll throw away this answer.)

Since 2 < x and x < 22, we get:  2 < x < 22

The calculator at this site will be able to give you this answer very quickly.

One meter is approximately 39.37 inches.

One yard is 36 inches.

Therefore, one meter is longer than one yard, and eight meters are longer than eight yards.

Let each short piece be  x cm  long.

Then, each longer piece will be  x + 3 cm long.

If you add the lengths of the three short piece to the two longer pieces, the total length will be 150 cm.

--->   (x) + (x) + (x) + )x + 3) + (x + 3)  =  150

Dropping the parentheses:

--->   x + x + x + x + 3 + x + 3  =  150

Adding the like terms:

--->   5x + 6  =  150

Subtracting 6 from both sides:

--->   5x  =  144

Dividing by 5:

--->   x  =  28.8 cm (for each short piece)

--->   x + 3  =  28.8 + 3  =  31.8 cm (for each longer piece)

The examples that you gave can be generalized to:  sin(x)  =  cos(90° - x).

log₄3 = x   --->   4^x = 3   --->   (2²)^x = 3   --->   2^2x = 3   --->   √(2^2x) = √3  

                --->   (2^2x)^1/2 = (3)^1/2   --->   2^x = (3)^1/2    --->   log₂(3)^1/2  = x

Therefore,   log₄3 =  log₂(3)^1/2