Question

1)Упрастить выражение:(а\а-1-2\а+1)\а"+1\а"+2а+1 2) (2\b-2-8\b"-4-1\b+2)*1\b"+4b+4 3)Решить неравенство: 3х"-2х-5>=0 4) (2-x)*(7x"-10х+3)<0 5)Решить уравнения: корень b+2=2+корень b-6 1g(x"+7)=2+1g(x-4) (log2x)"-3log2x+2=0 (x-1)*(x"-4)*x=0 6)Стороны треугольника абс совпадают с отрезками лежащими на прямых : 2х+у-7=0 х-3у+14=0 х+4у-14=0 Найти: координаты вершин треугольника. Периметр треугольника. cos A. Периметр треугольника АВС

Answer 1

Вот решение.Прости что долго)!"

Answer 2

To simplify the expressions and solve the equations and inequalities, let's go step by step:

1. Simplify the expression: (а\а-1-2\а+1)\а"+1\а"+2а+1

First, simplify the fractions within the parentheses: (а/(а-1) - 2/(а+1)) / (а + 1/(а+2) + 1)

To combine fractions with different denominators, find a common denominator, which is (а+1)(а+2):

[(а(а+2) - 2(а-1)) / ((а-1)(а+1)(а+2))] / [(а(а+2)(а+2) + (а+1)(а+2) + 1)]

Now, simplify further: [(а^2 + 2а - 2а + 2) / ((а-1)(а+1)(а+2))] / [(а^2 + 2а^2 + 2а + 2 + 1)]

Combine like terms: [(а^2 + 2) / ((а-1)(а+1)(а+2))] / [(3а^2 + 2а + 3)]

To divide fractions, invert the second fraction and multiply: [(а^2 + 2) / ((а-1)(а+1)(а+2))] * [(3а^2 + 2а + 3) / 1]

The result is the simplified expression.

2. Simplify the expression: (2\b-2-8\b"-4-1\b+2)*1\b"+4b+4

First, simplify the fractions within the parentheses: (2/b - 2 - 8/b - 4 - 1/b + 2) * 1/b + 4b + 4

Combine like terms in the parentheses: [(2 - 8 + 2 - b) / b] * [1/b + 4b + 4]

Simplify further: (-4 / b) * (1/b + 4b + 4)

Distribute the (-4 / b) to each term in the second parentheses: (-4 / b) * (1/b) + (-4 / b) * 4b + (-4 / b) * 4

The result is the simplified expression.

3. Solve the inequality: 3x - 2x - 5 >= 0

Combine like terms: x - 5 >= 0

Add 5 to both sides to isolate x: x >= 5

The solution for the inequality is x >= 5.

4. Solve the inequality: (2 - x)*(7x^2 - 10x + 3) < 0

First, factorize the quadratic expression: (2 - x)(7x^2 - 10x + 3) = -(x - 2)(7x - 3)*(x - 1)

Now, we have three critical points: x = 1, x = 2, and x = 3/7.

Create a number line and test intervals:

Interval 1: x < 1 Pick x = 0, substitute into the expression: -(0 - 2)(70 - 3)*(0 - 1) = -6 < 0 (True)

Interval 2: 1 < x < 2 Pick x = 1.5, substitute into the expression: -(1.5 - 2)(71.5 - 3)*(1.5 - 1) = 15/4 > 0 (False)

Interval 3: 2 < x < 3/7 Pick x = 0.5, substitute into the expression: -(0.5 - 2)(70.5 - 3)*(0.5 - 1) = -105/16 < 0 (True)

Interval 4: x > 3/7 Pick x = 1, substitute into the expression: -(1 - 2)(71 - 3)*(1 - 1) = 0 (False)

The solution to the inequality is 0 < x < 1 and x > 3/7.

5. Solve the equations: a) √(b + 2) = 2 + √(b - 6)

Square both sides to eliminate the square roots: b + 2 = 4 + 2√(b - 6) + (b - 6)

Combine like terms: 2√(b - 6) = 8

Divide by 2: √(b - 6) = 4

Square both sides again: b - 6 = 16

Add 6 to both sides: b = 22

The solution to the equation is b = 22.

b) log2x - 3log2x + 2 = 0

Combine the logarithms using the properties of logarithms: log2x - log2(x^3) + 2 = 0

Use the rule: log(a) - log(b) = log(a/b): log2(x/(x^3)) + 2 = 0

Simplify: log2(1/x^2) = -2

Now, rewrite in exponential form: 1/x^2 = 2^(-2)

Simplify: 1/x^2 = 1/4

Now, solve for x: x^2 = 4

Take the square root of both sides: x = ±2

So the solutions are x = 2 and x = -2.

c) (x - 1)*(x^2 - 4)*x = 0

First, find the roots of the equation by setting each factor to zero:

x - 1 = 0 ⇒ x = 1 x^2 - 4 = 0 ⇒ x^2 = 4 ⇒ x = ±2 x = 0

The solutions are x = 1, x = 2, x = -2, and x = 0.

6. Find the coordinates of the triangle's vertices, the triangle's perimeter, and cos A:

Given equations of the lines:

1. 2x + y - 7 = 0
2. x - 3y + 14 = 0
3. x + 4y - 14 = 0

To find the coordinates of the vertices, we need to find the points of intersection of these lines.

Solve equations (1) and (2) simultaneously: 2x + y = 7 x - 3y = -14

Multiply the second equation by 2 and add it to the first equation to eliminate x: 2x + y + 2x - 6y = 7 - 28 4x - 5y = -21

Solve for x: 4x = 5y - 21 x = (5y - 21)/4

Substitute x into equation (2) and solve for y: (5y - 21)/4 - 3y = -14 5y - 21 - 12y = -56 -7y = -35 y =