Mister Exam
((15^x-3^(x+1)-5^(x+1)+15)*1/(-x^2+2*x))>=0 inequation
The solution
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x x + 1 x + 1
15 - 3 - 5 + 15
-------------------------- >= 0
2
- x + 2*x
$$\frac{\left(- 5^{x + 1} + \left(15^{x} - 3^{x + 1}\right)\right) + 15}{- x^{2} + 2 x} \geq 0$$
(-5^(x + 1) + 15^x - 3^(x + 1) + 15)/(-x^2 + 2*x) >= 0
Detail solution
Given the inequality:
$$\frac{\left(- 5^{x + 1} + \left(15^{x} - 3^{x + 1}\right)\right) + 15}{- x^{2} + 2 x} \geq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{\left(- 5^{x + 1} + \left(15^{x} - 3^{x + 1}\right)\right) + 15}{- x^{2} + 2 x} = 0$$
Solve:
$$x_{1} = 0.682606194485985$$
$$x_{2} = 1.46497352071793$$
$$x_{1} = 0.682606194485985$$
$$x_{2} = 1.46497352071793$$
This roots
$$x_{1} = 0.682606194485985$$
$$x_{2} = 1.46497352071793$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 0.682606194485985$$
=
$$0.582606194485985$$
substitute to the expression
$$\frac{\left(- 5^{x + 1} + \left(15^{x} - 3^{x + 1}\right)\right) + 15}{- x^{2} + 2 x} \geq 0$$
$$\frac{\left(- 5^{0.582606194485985 + 1} + \left(- 3^{0.582606194485985 + 1} + 15^{0.582606194485985}\right)\right) + 15}{- 0.582606194485985^{2} + 0.582606194485985 \cdot 2} \geq 0$$
one of the solutions of our inequality is:
$$x \leq 0.682606194485985$$
Other solutions will get with the changeover to the next point
etc.
The answer:
$$x \leq 0.682606194485985$$
$$x \geq 1.46497352071793$$
$$\frac{\left(- 5^{x + 1} + \left(15^{x} - 3^{x + 1}\right)\right) + 15}{- x^{2} + 2 x} \geq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{\left(- 5^{x + 1} + \left(15^{x} - 3^{x + 1}\right)\right) + 15}{- x^{2} + 2 x} = 0$$
Solve:
$$x_{1} = 0.682606194485985$$
$$x_{2} = 1.46497352071793$$
$$x_{1} = 0.682606194485985$$
$$x_{2} = 1.46497352071793$$
This roots
$$x_{1} = 0.682606194485985$$
$$x_{2} = 1.46497352071793$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 0.682606194485985$$
=
$$0.582606194485985$$
substitute to the expression
$$\frac{\left(- 5^{x + 1} + \left(15^{x} - 3^{x + 1}\right)\right) + 15}{- x^{2} + 2 x} \geq 0$$
$$\frac{\left(- 5^{0.582606194485985 + 1} + \left(- 3^{0.582606194485985 + 1} + 15^{0.582606194485985}\right)\right) + 15}{- 0.582606194485985^{2} + 0.582606194485985 \cdot 2} \geq 0$$
1.67605755234356 >= 0
one of the solutions of our inequality is:
$$x \leq 0.682606194485985$$
_____ _____
\ /
-------•-------•-------
x1 x2Other solutions will get with the changeover to the next point
etc.
The answer:
$$x \leq 0.682606194485985$$
$$x \geq 1.46497352071793$$
Solving inequality on a graph