Q.1the set of all solutions of the inequality (1/2)^((x^2)-2x)< (1/4) contains the set (3,infinite). But how?

1/2)^z<(1/2)^2

take the log (base 2 ) of each side

-z<-2
z>2
solve for x (z=x^2-2x)
x^2-2x-2>0

x=(2+-sqrt(4+8))/2=1+-sqrt3
so test these two roots, and you see that only 1+sqrt3 will satisfy the inequality.
x>2.73

Great solution,seen 14 years after this answer has been posted here! Hope you'll be doin well

To determine the set of solutions for the inequality (1/2)^((x^2)-2x) < (1/4), we can start by simplifying the inequality:

(1/2)^((x^2)-2x) < (1/4)

First, let's simplify the right side of the inequality by writing (1/4) as (1/2)^2:

(1/2)^((x^2)-2x) < (1/2)^2

Now, we have the same base on both sides of the inequality, so we can equate the exponents:

((x^2)-2x) < 2

Next, let's solve the inequality using traditional algebraic methods. We'll start by moving all terms to one side to get a quadratic inequality:

(x^2) - 2x - 2 < 0

Now, we can factor the quadratic equation:

(x - 3)(x + 1) < 0

Next, we need to determine the sign of the expression (x - 3)(x + 1) in order to find the intervals where the inequality is true. We can use a sign chart or test values within each interval.

1. If x < -1:
Testing a value, like -2, in the inequality:
(-2 - 3)(-2 + 1) = (-5)(-1) = 5 > 0 (not less than zero).

2. If -1 < x < 3:
Testing a value, like 0, in the inequality:
(0 - 3)(0 + 1) = (-3)(1) = -3 < 0

3. If x > 3:
Testing a value, like 4, in the inequality:
(4 - 3)(4 + 1) = (1)(5) = 5 > 0 (not less than zero).

Hence, the solution to the inequality (1/2)^((x^2)-2x) < (1/4) is -1 < x < 3.

Therefore, the set of all solutions of the inequality is the interval (-1, 3). Please note that this interval does not include the values -1 and 3 since we have a strict inequality, <, and not ≤.

To solve the inequality (1/2)^((x^2)-2x) < (1/4), we can follow these steps:

Step 1: Set the expression inside the inequality to be equal to zero to find the critical points.
(1/2)^((x^2)-2x) = (1/4)

Step 2: Simplify and rewrite the equation.
2^((x^2)-2x) = 4

Step 3: Rewrite 4 as 2^2.
2^((x^2)-2x) = 2^2

Step 4: Set the exponents equal to each other.
(x^2) - 2x = 2

Step 5: Rearrange the equation to form a quadratic equation.
x^2 - 2x - 2 = 0

Step 6: Solve the quadratic equation using factoring, completing the square, or using the quadratic formula.
By factoring, we can rewrite the equation as:
(x - 2)(x + 1) = 0

Therefore, x can be either 2 or -1.

Step 7: Plot these critical points on a number line.

-∞ -1 2 +∞
|_____|------|_____|______|

Step 8: Check the signs of the inequality within each interval.

For x < -1:
Choose a value less than -1, e.g., x = -2.
When we substitute x = -2 into the inequality, we get:
(1/2)^((-2^2)-2(-2)) < (1/4)
(1/2)^(4+4) < (1/4)
(1/2)^8 < (1/4)
1/256 < 1/4, which is true.

For -1 < x < 2:
Choose a value between -1 and 2, e.g., x = 0.
When we substitute x = 0 into the inequality, we get:
(1/2)^((0^2)-2(0)) < (1/4)
(1/2)^(0-0) < (1/4)
(1/2)^0 < (1/4)
1 < 1/4, which is false.

For x > 2:
Choose a value greater than 2, e.g., x = 3.
When we substitute x = 3 into the inequality, we get:
(1/2)^((3^2)-2(3)) < (1/4)
(1/2)^(9-6) < (1/4)
(1/2)^3 < (1/4)
1/8 < 1/4, which is true.

Step 9: Identify the intervals satisfying the inequality.
From the analysis above, we can see that the inequality is true for x < -1 and x > 2. Combining these intervals, we can say the set of all solutions of the inequality is (-∞, -1) U (2, +∞).

Therefore, the set of all solutions of the inequality (1/2)^((x^2)-2x) < (1/4) is (-∞, -1) U (2, +∞), which contains the set (3, +∞).