Mister Exam
(4*x-7)*log(_x)^2-4*x+5*(3*x-5)>=0 inequation
The solution
You have entered
[src]
2 (4*x - 7)*log (x) - 4*x + 5*(3*x - 5) >= 0
$$\left(- 4 x + \left(4 x - 7\right) \log{\left(x \right)}^{2}\right) + 5 \left(3 x - 5\right) \geq 0$$
-4*x + (4*x - 7)*log(x)^2 + 5*(3*x - 5) >= 0
Detail solution
Given the inequality:
$$\left(- 4 x + \left(4 x - 7\right) \log{\left(x \right)}^{2}\right) + 5 \left(3 x - 5\right) \geq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(- 4 x + \left(4 x - 7\right) \log{\left(x \right)}^{2}\right) + 5 \left(3 x - 5\right) = 0$$
Solve:
$$x_{1} = -0.239750193867111 + 0.897290993519563 i$$
$$x_{2} = 2.17831875631327$$
Exclude the complex solutions:
$$x_{1} = 2.17831875631327$$
This roots
$$x_{1} = 2.17831875631327$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 2.17831875631327$$
=
$$2.07831875631327$$
substitute to the expression
$$\left(- 4 x + \left(4 x - 7\right) \log{\left(x \right)}^{2}\right) + 5 \left(3 x - 5\right) \geq 0$$
$$\left(- 2.07831875631327 \cdot 4 + \left(-7 + 2.07831875631327 \cdot 4\right) \log{\left(2.07831875631327 \right)}^{2}\right) + 5 \left(-5 + 2.07831875631327 \cdot 3\right) \geq 0$$
but
Then
$$x \leq 2.17831875631327$$
no execute
the solution of our inequality is:
$$x \geq 2.17831875631327$$
$$\left(- 4 x + \left(4 x - 7\right) \log{\left(x \right)}^{2}\right) + 5 \left(3 x - 5\right) \geq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(- 4 x + \left(4 x - 7\right) \log{\left(x \right)}^{2}\right) + 5 \left(3 x - 5\right) = 0$$
Solve:
$$x_{1} = -0.239750193867111 + 0.897290993519563 i$$
$$x_{2} = 2.17831875631327$$
Exclude the complex solutions:
$$x_{1} = 2.17831875631327$$
This roots
$$x_{1} = 2.17831875631327$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 2.17831875631327$$
=
$$2.07831875631327$$
substitute to the expression
$$\left(- 4 x + \left(4 x - 7\right) \log{\left(x \right)}^{2}\right) + 5 \left(3 x - 5\right) \geq 0$$
$$\left(- 2.07831875631327 \cdot 4 + \left(-7 + 2.07831875631327 \cdot 4\right) \log{\left(2.07831875631327 \right)}^{2}\right) + 5 \left(-5 + 2.07831875631327 \cdot 3\right) \geq 0$$
-1.43565649834005 >= 0
but
-1.43565649834005 < 0
Then
$$x \leq 2.17831875631327$$
no execute
the solution of our inequality is:
$$x \geq 2.17831875631327$$
_____
/
-------•-------
x1
Solving inequality on a graph