Question

Розв'язати нерівності 1)4≥ 16^x+12)(√7)^х ≤ 1/493)2^х^2 - 9х + 17,5 > √2/84) (3/2)^х < -35) (0,25)^4-х ≤ 16/2^х-26) 4^х - 6*2^х-1 >​ 4

Answer 1

Объяснение:

решение во вложении......

Answer 2

1. 4 ≥ (16^x)(7^(x)) ≤ 1/49

To solve this inequality, first notice that 4 and 1/49 are powers of 2 (2^2 and 1/7^2). We can rewrite the inequality as:

2^2 ≥ (2^4)^x * (7^x) ≥ (1/7^2)

Now, we can use the properties of exponents to simplify the inequality:

2^2 ≥ 2^(4x) * 7^x ≥ 1/7^2

Now, we can compare the exponents and solve for x:

2 ≥ 4x 1/7^2 ≤ 7^x

1. x ≤ 1/2
2. x ≥ -2

So, the solution to the inequality is: -2 ≤ x ≤ 1/2

2. 2^x^2 - 9x + 17.5 > √2/8

To solve this inequality, you can first simplify it:

2^x^2 - 9x + 17.5 > √2/8

Now, let's convert √2/8 into its exponential form:

√2/8 = 2^(1/2) / 2^3 = 2^((1/2) - 3) = 2^(-5/2)

So, the inequality becomes:

2^x^2 - 9x + 17.5 > 2^(-5/2)

Now, you can set both sides of the inequality with the same base and compare exponents:

2^x^2 - 9x + 17.5 > 2^(-5/2)

x^2 - (9/2)x + 17.5 > -5/2

x^2 - (9/2)x + 17.5 + 5/2 > 0

x^2 - (9/2)x + 11 > 0

To solve this quadratic inequality, you can use the quadratic formula:

x = [9/2 ± √((9/2)^2 - 4111)] / (2*1)

x = [9/2 ± √(81/4 - 44)] / 2

x = [9/2 ± √(81/4 - 176/4)] / 2

x = [9/2 ± √(-95/4)] / 2

Since the discriminant is negative, there are no real solutions to this quadratic inequality. Therefore, the inequality has no solution in real numbers.

3. (3/2)^x < -3

To solve this inequality, we can rewrite it as:

(2/3)^x > 3

Now, let's take the natural logarithm (ln) of both sides of the inequality to solve for x:

ln((2/3)^x) > ln(3)

Using the properties of logarithms, we can move the exponent x down:

x * ln(2/3) > ln(3)

Now, divide both sides by ln(2/3):

x > ln(3) / ln(2/3)

You can calculate the approximate value of x using a calculator:

x > -1.63093

So, the solution to the inequality is x > -1.63093.

4. (0.25)^(4-x) ≤ 16 / 2^x-2

To solve this inequality, first, simplify the right side:

(0.25)^(4-x) ≤ 16 / (2^(x-2))

Now, rewrite 16 as a power of 2 (2^4):

(0.25)^(4-x) ≤ (2^4) / (2^(x-2))

Using the properties of exponents, we can rewrite the right side:

(0.25)^(4-x) ≤ 2^(4 - (x-2))

Simplify further:

(0.25)^(4-x) ≤ 2^(6-x)

Now, both sides have the same base (0.25 and 2), so we can compare exponents:

4-x ≤ 6-x

Now, subtract 4-x from both sides:

0 ≤ 2

Since 0 is always less than or equal to 2, the inequality is true for all values of x.

5. 4^x - 6 * 2^(x-1) > 4

First, let's notice that 4 is also a power of 2 (2^2). We can rewrite the inequality as:

2^(2x) - 6 * 2^(x-1) > 2^2

Now, we can use properties of exponents:

2^(2x) - 6 * (2^x/2) > 2^2

Simplify further:

2^(2x) - 3 * 2^x > 2^2

Now, let's introduce a substitution. Let y = 2^x. The inequality becomes:

y^2 - 3y > 4

Rearrange the inequality:

y^2 - 3y - 4 > 0

Now, you can factor the left side of the inequality:

(y - 4)(y + 1) > 0

Now, find the values of y that make the inequality true by considering the signs of each factor:

1. y - 4 > 0 => y > 4
2. y + 1 > 0 => y > -1

So, the solution is y > 4 and y > -1. Since y = 2^x, the solution in terms of x is:

2^x > 4 and 2^x > -1

The condition 2^x > -1 always holds true, so we only need to consider 2^x > 4:

2^x > 4

To solve this, take the logarithm of both sides:

x * ln(2) > ln(4)

x * 0.6931 > 1.3863

x > 1.3863 / 0.6931

x > 2

So, the solution to the inequality is x > 2.