Algebra precalculus – solving two equations in three variables

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Algebra Pre-calculus – However, the attempt to solve this inequality has not progressed at all.

The problem is, that $$x, y, z$$ are right breaks, and each of them is greater than zero.

Given: $$x + y + z = 2$$

Must prove $$frac {x. y. z} {(1-x). (1-y). (1-z)} 8q$$,

I tried to solve this with AM $$geq$$ GM inequality.

Attempt :

$$frac { frac {1-x} {x} + frac {1-y} {y} + frac {1-z} {z}} {3}$$ $$geq$$ $$biggl ( frac {(1-x). (1-y). (1-z)} {x.y.z} biggl) ^ {1/3}$$

What should I do to calculate the value of? $$frac {1} {x} + frac {1} {y} + frac {1} {z}$$ ?

Algebra precalculus – $x_1 x_2 x_3 x_4 + x_2 x_3 x_4 x_5 + …… + x_n x_1 x_2 x_3 = 0$ What is $n$?

It's really hard for me to understand this problem. What I understood is that I have to find a natural number $$n$$ for which the equation holds, whatever the values ​​of $$x_i$$ That's impossible. Because of this, some conditions seem to be on $$x_i$$ s are necessary ..

Algebra Pre-calculus – Problems in interpreting information to develop mathematical functions

Context: assignment of the university code

So far I've developed two models based on this information:

For the surface zone I have $$T (l) = frac {-11} {45} l + 24$$ and for the deep zone I have $$T (l) = 2$$ from where $$T$$ is the summer temperature in degrees Celsius and $$l$$ is the latitude in degrees (where $$0$$ is the equator and $$90$$ are the poles).

I have trouble interpreting this information to develop a linear model for the summer temperature of seawater ($$T$$) in the thermocline zone. I know that this model will be different from the last two, because it depends on depth (not latitude), but the lack of data really makes me nervous.

Any guidance would be very grateful.

Algebra Pre-Calculus – How to multiply both sides of $frac {5} {X_1-X_2}> 10$ with $X_1-X_2$ if $X_i$ are independent of each other as random variables?

Suppose we have random variables $$X_1$$ and $$X_2$$ they are independent and distributed identically.

Suppose I am interested in the inequality $$frac {5} {X_1-X_2}> 10$$,

How can I multiply both sides of this inequality? $$X_1-X_2$$

Especially since $$X_1$$ and $$X_2$$ are random variables, I do not know if $$X_1 -X_2$$ is positive or negative. So I do not know if I have to reverse the sign of inequality or not.

Furthermore, a statement would please $$frac {5} {X_1-X_2}> 10$$ THEN AND ONLY THEN, IF $$. 5> X_1-X_2$$ be right?

I think it would be just an attempt to prove it. Suppose that $$.5> X_1-X_2$$, Then it has to be that $$X_1> X_2$$ and we can rearrange to get $$frac {5} {X_1-X_2}> 10$$ by $$X_1-X_2$$ , (So the IF direction applies)

And for the only if directionagain if $$frac {5} {X_1-X_2}> 10$$ by $$X_1-X_2$$ then $$X_1-X_2$$ must be positive, since a negative number can not be greater than $$10$$and we get the result by rearranging.

Algebra Precalculus – Number of possible integer values ​​of x for which a given expression is an integer

How many integers, $$x$$Check the following

$$begin {equation *} frac {x ^ 3 + 2x ^ 2 + 9} {x ^ 2 + 4x + 5} end {equation *}$$

is an integer?

I did it:

$$begin {equation *} frac {x ^ 3 + 2x ^ 2 + 9} {x ^ 2 + 4x + 5} = x-2 + frac {3x + 19} {x ^ 2 + 4x + 5} end {equation *}$$

but I can not continue.

Algebra precalculus – Complex number: $frac {(3 + i) ^ 2} {(1 + 2i) ^ 2}$ – can not access the textbook solution

I have a complex quotient $$frac {(3 + i) ^ 2} {(1 + 2i) ^ 2}$$

The solution in my textbook is $$-2i$$, I came to different solutions and would like to know where I went wrong.

So far I've been working with the complex number i in my textbook chapter ($$sqrt {-1}$$).

I understand that one can not divide by a complex number in the denominator, so I have to multiply both the numerator and the denominator with the complex conjugate.

However, I am confused in this exercise because my expression is nested in parentheses and is square. For example, if my denominator was straight $$1 + 2i$$ I know that would be the complex conjugate $$1-2i$$,

So I'm confused about what to do because the whole denominator is in parentheses and square.

Only with what I know did I try to solve the square in numerator and denominator:

$$(3 + i) ^ 2$$ = $$3 ^ 2 + i ^ 2$$ = $$9-1$$ = $$8$$

For the denominator:
$$(1 + 2i) ^ 2$$ = $$1 ^ 2 + 2 ^ 2i ^ 2$$ = $$1 + 4 * -1$$ = $$1-4$$ = $$-3$$

Then I would arrive $$frac {8} {- 3}$$ That's not the solution.

How do I arrive? $$-2i$$?

Algebra precalculus – Find the extremes of $cos left ( frac pi2 cos x right) + cos left ( frac pi2 sin x right)$ without distinction

The question is: find minimum and maximum of $$f (x)$$:
$$f (x) = cos left ( frac pi2 cos x right) + cos left ( frac pi2 sin x right)$$
without distinction.

This problem should only be solved with pre-calculation knowledge, but I have no idea how to do it.

$$f (x)$$ decreases monotonously $$frac {n pi} 2$$ to $$frac {n pi} 2+ frac pi4$$and rises monotonously $$frac {n pi} 2+ frac pi4$$ to $$frac {(n + 1) pi} 2$$But how can one prove the monotony without calculation?

I also tried to transform the expression into
begin {align} f (x) = & 2 cos left ( frac pi4 ( cos x + sin x) right) cos left ( frac pi4 ( cos x- sin x ) right) \ = & 2 cos left ( frac { sqrt2 pi} 4 sin left (x + frac pi4 right) right) cos left ( frac { sqrt2 pi} 4 sin left (-x + frac pi4 right) right) end {align}
but found it has the same problem to prove the monotony.

Algebra precalculus – What extension do I use for ln (x) if I do not know what x is greater or less than?

I try to make extensions for ln and log in to honor Algebra 2, but I can not figure that out. The equation I'm working on is ln15x. I came to ln15 + lnx, but I do not know how to simplify it. Note: This is probably a very simple concept, but I've been sick for a few months, so I have no idea what I'm mainly doing and the internet is not helping.

Algebra precalculus – Solve the equation $13x + 2 (3x + 2) sqrt {x + 3} + 42 = 0$.

Solve the equation $$13x + 2 (3x + 2) sqrt {x + 3} + 42 = 0$$,

To let $$y = sqrt {x + 3} implies 3 = y ^ 2 – x$$,

large begin {align} & 13x + 2 (3x + 2) sqrt {x + 3} +42 \ = & 14 (x + 3) + (6x + 4) y – x \ = & 14y ^ 2 + [6(x + 3) – 14]y – x = & 14y (y – 1) – (y ^ 2 – x – 9) y ^ 3 – x \ = & 14y (y – 1) + x (y ^ 3 – y) – y ^ 3 (y ^ 2 – 1) + 8y ^ 3 \ = & 14y (y – 1) + (xy ^ 2 + xy + x) (y – 1) – (y ^ 4 + y ^ 3) (y – 1) + 8y ^ 3 \ = & (- y ^ 4 + y ^ 3 + xy ^ 2 + xy + x + 14y) (y – 1) + 8y ^ 3 \ end {align}

And I'm stuck here.