Mister Exam
(3*x^2-17*x+18)/(x^2-5*x+4)<2 inequation
The solution
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2 3*x - 17*x + 18 ---------------- < 2 2 x - 5*x + 4
$$\frac{\left(3 x^{2} - 17 x\right) + 18}{\left(x^{2} - 5 x\right) + 4} < 2$$
(3*x^2 - 17*x + 18)/(x^2 - 5*x + 4) < 2
Detail solution
Given the inequality:
$$\frac{\left(3 x^{2} - 17 x\right) + 18}{\left(x^{2} - 5 x\right) + 4} < 2$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{\left(3 x^{2} - 17 x\right) + 18}{\left(x^{2} - 5 x\right) + 4} = 2$$
Solve:
Given the equation:
$$\frac{\left(3 x^{2} - 17 x\right) + 18}{\left(x^{2} - 5 x\right) + 4} = 2$$
Multiply the equation sides by the denominators:
4 + x^2 - 5*x
we get:
$$\frac{\left(\left(3 x^{2} - 17 x\right) + 18\right) \left(x^{2} - 5 x + 4\right)}{\left(x^{2} - 5 x\right) + 4} = 2 x^{2} - 10 x + 8$$
$$3 x^{2} - 17 x + 18 = 2 x^{2} - 10 x + 8$$
Move right part of the equation to
left part with negative sign.
The equation is transformed from
$$3 x^{2} - 17 x + 18 = 2 x^{2} - 10 x + 8$$
to
$$x^{2} - 7 x + 10 = 0$$
This equation is of the form
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 1$$
$$b = -7$$
$$c = 10$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = 5$$
$$x_{2} = 2$$
$$x_{1} = 5$$
$$x_{2} = 2$$
$$x_{1} = 5$$
$$x_{2} = 2$$
This roots
$$x_{2} = 2$$
$$x_{1} = 5$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{2}$$
For example, let's take the point
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 2$$
=
$$\frac{19}{10}$$
substitute to the expression
$$\frac{\left(3 x^{2} - 17 x\right) + 18}{\left(x^{2} - 5 x\right) + 4} < 2$$
$$\frac{\left(- \frac{17 \cdot 19}{10} + 3 \left(\frac{19}{10}\right)^{2}\right) + 18}{\left(- \frac{5 \cdot 19}{10} + \left(\frac{19}{10}\right)^{2}\right) + 4} < 2$$
one of the solutions of our inequality is:
$$x < 2$$
Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < 2$$
$$x > 5$$
$$\frac{\left(3 x^{2} - 17 x\right) + 18}{\left(x^{2} - 5 x\right) + 4} < 2$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{\left(3 x^{2} - 17 x\right) + 18}{\left(x^{2} - 5 x\right) + 4} = 2$$
Solve:
Given the equation:
$$\frac{\left(3 x^{2} - 17 x\right) + 18}{\left(x^{2} - 5 x\right) + 4} = 2$$
Multiply the equation sides by the denominators:
4 + x^2 - 5*x
we get:
$$\frac{\left(\left(3 x^{2} - 17 x\right) + 18\right) \left(x^{2} - 5 x + 4\right)}{\left(x^{2} - 5 x\right) + 4} = 2 x^{2} - 10 x + 8$$
$$3 x^{2} - 17 x + 18 = 2 x^{2} - 10 x + 8$$
Move right part of the equation to
left part with negative sign.
The equation is transformed from
$$3 x^{2} - 17 x + 18 = 2 x^{2} - 10 x + 8$$
to
$$x^{2} - 7 x + 10 = 0$$
This equation is of the form
a*x^2 + b*x + c = 0
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 1$$
$$b = -7$$
$$c = 10$$
, then
D = b^2 - 4 * a * c =
(-7)^2 - 4 * (1) * (10) = 9
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = 5$$
$$x_{2} = 2$$
$$x_{1} = 5$$
$$x_{2} = 2$$
$$x_{1} = 5$$
$$x_{2} = 2$$
This roots
$$x_{2} = 2$$
$$x_{1} = 5$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{2}$$
For example, let's take the point
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 2$$
=
$$\frac{19}{10}$$
substitute to the expression
$$\frac{\left(3 x^{2} - 17 x\right) + 18}{\left(x^{2} - 5 x\right) + 4} < 2$$
$$\frac{\left(- \frac{17 \cdot 19}{10} + 3 \left(\frac{19}{10}\right)^{2}\right) + 18}{\left(- \frac{5 \cdot 19}{10} + \left(\frac{19}{10}\right)^{2}\right) + 4} < 2$$
347 --- < 2 189
one of the solutions of our inequality is:
$$x < 2$$
_____ _____
\ /
-------ο-------ο-------
x2 x1Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < 2$$
$$x > 5$$
Solving inequality on a graph
Rapid solution
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Or(And(1 < x, x < 2), And(4 < x, x < 5))
$$\left(1 < x \wedge x < 2\right) \vee \left(4 < x \wedge x < 5\right)$$
((1 < x)∧(x < 2))∨((4 < x)∧(x < 5))
Rapid solution 2
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(1, 2) U (4, 5)
$$x\ in\ \left(1, 2\right) \cup \left(4, 5\right)$$
x in Union(Interval.open(1, 2), Interval.open(4, 5))