Mister Exam
20^x-64*5^x-4^x+64<=0 inequation
The solution
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x x x 20 - 64*5 - 4 + 64 <= 0
$$\left(- 4^{x} + \left(20^{x} - 64 \cdot 5^{x}\right)\right) + 64 \leq 0$$
-4^x + 20^x - 64*5^x + 64 <= 0
Detail solution
Given the inequality:
$$\left(- 4^{x} + \left(20^{x} - 64 \cdot 5^{x}\right)\right) + 64 \leq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(- 4^{x} + \left(20^{x} - 64 \cdot 5^{x}\right)\right) + 64 = 0$$
Solve:
$$x_{1} = 0$$
$$x_{2} = 3$$
$$x_{1} = 0$$
$$x_{2} = 3$$
This roots
$$x_{1} = 0$$
$$x_{2} = 3$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 0$$
=
$$-0.1$$
substitute to the expression
$$\left(- 4^{x} + \left(20^{x} - 64 \cdot 5^{x}\right)\right) + 64 \leq 0$$
$$\left(\left(- \frac{64}{5^{0.1}} + 20^{-0.1}\right) - 4^{-0.1}\right) + 64 \leq 0$$
but
Then
$$x \leq 0$$
no execute
one of the solutions of our inequality is:
$$x \geq 0 \wedge x \leq 3$$
$$\left(- 4^{x} + \left(20^{x} - 64 \cdot 5^{x}\right)\right) + 64 \leq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(- 4^{x} + \left(20^{x} - 64 \cdot 5^{x}\right)\right) + 64 = 0$$
Solve:
$$x_{1} = 0$$
$$x_{2} = 3$$
$$x_{1} = 0$$
$$x_{2} = 3$$
This roots
$$x_{1} = 0$$
$$x_{2} = 3$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 0$$
=
$$-0.1$$
substitute to the expression
$$\left(- 4^{x} + \left(20^{x} - 64 \cdot 5^{x}\right)\right) + 64 \leq 0$$
$$\left(\left(- \frac{64}{5^{0.1}} + 20^{-0.1}\right) - 4^{-0.1}\right) + 64 \leq 0$$
9.38482884448061 <= 0
but
9.38482884448061 >= 0
Then
$$x \leq 0$$
no execute
one of the solutions of our inequality is:
$$x \geq 0 \wedge x \leq 3$$
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