Mister Exam
16^(x^2+2x+10)-0.25^(2x^2+4x-116)<=0 inequation
The solution
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2 2 x + 2*x + 10 116 - 4*x - 2*x 16 - 4 <= 0
$$- \left(\frac{1}{4}\right)^{\left(2 x^{2} + 4 x\right) - 116} + 16^{\left(x^{2} + 2 x\right) + 10} \leq 0$$
-(1/4)^(2*x^2 + 4*x - 116) + 16^(x^2 + 2*x + 10) <= 0
Detail solution
Given the inequality:
$$- \left(\frac{1}{4}\right)^{\left(2 x^{2} + 4 x\right) - 116} + 16^{\left(x^{2} + 2 x\right) + 10} \leq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$- \left(\frac{1}{4}\right)^{\left(2 x^{2} + 4 x\right) - 116} + 16^{\left(x^{2} + 2 x\right) + 10} = 0$$
Solve:
$$x_{1} = -6$$
$$x_{2} = 4$$
$$x_{1} = -6$$
$$x_{2} = 4$$
This roots
$$x_{1} = -6$$
$$x_{2} = 4$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$-6 + - \frac{1}{10}$$
=
$$- \frac{61}{10}$$
substitute to the expression
$$- \left(\frac{1}{4}\right)^{\left(2 x^{2} + 4 x\right) - 116} + 16^{\left(x^{2} + 2 x\right) + 10} \leq 0$$
$$- \left(\frac{1}{4}\right)^{-116 + \left(\frac{\left(-61\right) 4}{10} + 2 \left(- \frac{61}{10}\right)^{2}\right)} + 16^{10 + \left(\frac{\left(-61\right) 2}{10} + \left(- \frac{61}{10}\right)^{2}\right)} \leq 0$$
but
Then
$$x \leq -6$$
no execute
one of the solutions of our inequality is:
$$x \geq -6 \wedge x \leq 4$$
$$- \left(\frac{1}{4}\right)^{\left(2 x^{2} + 4 x\right) - 116} + 16^{\left(x^{2} + 2 x\right) + 10} \leq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$- \left(\frac{1}{4}\right)^{\left(2 x^{2} + 4 x\right) - 116} + 16^{\left(x^{2} + 2 x\right) + 10} = 0$$
Solve:
$$x_{1} = -6$$
$$x_{2} = 4$$
$$x_{1} = -6$$
$$x_{2} = 4$$
This roots
$$x_{1} = -6$$
$$x_{2} = 4$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$-6 + - \frac{1}{10}$$
=
$$- \frac{61}{10}$$
substitute to the expression
$$- \left(\frac{1}{4}\right)^{\left(2 x^{2} + 4 x\right) - 116} + 16^{\left(x^{2} + 2 x\right) + 10} \leq 0$$
$$- \left(\frac{1}{4}\right)^{-116 + \left(\frac{\left(-61\right) 4}{10} + 2 \left(- \frac{61}{10}\right)^{2}\right)} + 16^{10 + \left(\frac{\left(-61\right) 2}{10} + \left(- \frac{61}{10}\right)^{2}\right)} \leq 0$$
24
--
25 25___ <= 0
- 2722258935367507707706996859454145691648*2 + 1393796574908163946345982392040522594123776*\/ 2
but
24
--
25 25___ >= 0
- 2722258935367507707706996859454145691648*2 + 1393796574908163946345982392040522594123776*\/ 2
Then
$$x \leq -6$$
no execute
one of the solutions of our inequality is:
$$x \geq -6 \wedge x \leq 4$$
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