Question

1.Неравенство log5(2x-1)<22.Уравнение tg(x-п/4)=корень из 33.Середина отрезка АВ с концами в точках А(-2;3;5) и В(2;-3;7) принадлежит:оси х, оси у, оси z, площади ху?4.Найдите область значения функции f(x)=3-корень из х5.уравнение 1/8 корень 2^x-1 =4^-1.25Помогите пожалуйста

Answer 1

1) $\left \{ {{2x-1<25} \atop {2x-1>0}} \right.$
2x-1<25      2x-1>0
2x<26          x>0.5
x<13
x∈ (0.5;13)
Ответ:  (0,5;13)
2) tg(x-π/4) = √3
x-π/4 = π/3 + πn
x= 7π/12 +πn
Ответ: 7π/12 + πn
4) f(x)  =3-√x
√x ≥ 0
-беск. < 3-√x ≤3
Ответ: (-беск.; 3]
5) $\frac{ \sqrt{ 2^{x-1} } }{8} = 4^{-1.25}$
$\sqrt{ 2^{x-1} } = 2^{-2.5} * 2^{3}$
$\sqrt{ 2^{x-1} }= 2^{0.5}$
$2^{x-1} = 2$
x-1 = 1
x=2
3) C (с₁;c₂;c₃) - середина отрезка AB
c₁ = (a₁+b₁) / 2 =(-2+2)/2 = 0
c₂ = (3-3)/2 = 0
c₃ = (2+7)/2 = 6
Ответ: середина отрезка АВ принадлежит оси z

Answer 2

1. Неравенство log5(2x-1) < 2

To solve the inequality log5(2x-1) < 2, we can rewrite it as an exponential equation. The base of the logarithm is 5, so we can rewrite the inequality as:

5^(log5(2x-1)) < 5^2

Simplifying further:

2x - 1 < 25

Adding 1 to both sides:

2x < 26

Dividing both sides by 2:

x < 13

Therefore, the solution to the inequality log5(2x-1) < 2 is x < 13.

2. Уравнение tg(x-π/4) = √33

To solve the equation tg(x-π/4) = √33, we can take the arctangent of both sides to isolate x:

x - π/4 = arctan(√33)

Adding π/4 to both sides:

x = arctan(√33) + π/4

Therefore, the solution to the equation tg(x-π/4) = √33 is x = arctan(√33) + π/4.

3. Середина отрезка АВ с концами в точках А(-2;3;5) и В(2;-3;7) принадлежит: оси х, оси у, оси z, площади ху?

To determine where the midpoint of the line segment AB, with endpoints A(-2,3,5) and B(2,-3,7), lies, we can calculate the coordinates of the midpoint.

The midpoint of a line segment with endpoints (x1, y1, z1) and (x2, y2, z2) is given by the formula:

Midpoint = ((x1 + x2)/2, (y1 + y2)/2, (z1 + z2)/2)

Using the coordinates of points A and B:

Midpoint = ((-2 + 2)/2, (3 + (-3))/2, (5 + 7)/2) = (0, 0, 6/2) = (0, 0, 3)

Therefore, the midpoint of the line segment AB lies on the z-axis.

4. Найдите область значений функции f(x) = 3 - √x

To find the range of the function f(x) = 3 - √x, we need to determine the possible values that f(x) can take.

The square root function (√x) is defined for non-negative real numbers. Therefore, the expression √x is only defined for x ≥ 0.

Since f(x) = 3 - √x, the range of f(x) will be all real numbers less than or equal to 3.

Therefore, the range of the function f(x) = 3 - √x is (-∞, 3].

5. Уравнение 1/8 * √(2^x-1) = 4^(-1.25)

To solve the equation 1/8 * √(2^x-1) = 4^(-1.25), we can start by simplifying the equation.

First, let's simplify the right side of the equation:

4^(-1.25) = 1/4^(1.25) = 1/√4 = 1/2

Now, let's rewrite the equation:

1/8 * √(2^x-1) = 1/2

Multiplying both sides of the equation by 8:

√(2^x-1) = 4

Squaring both sides of the equation:

2^x - 1 = 16

Adding 1 to both sides:

2^x = 17

Taking the logarithm base 2 of both sides:

x = log2(17)

Therefore, the solution to the equation 1/8 * √(2^x-1) = 4^(-1.25) is x = log2(17). I hope this helps! Let me know if you have any further questions.