Mister Exam
(x+3)*(4-x)*(2x+5)/(3x+1)*(x+4)>0 inequation
The solution
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(x + 3)*(4 - x)*(2*x + 5)
-------------------------*(x + 4) > 0
3*x + 1
$$\frac{\left(4 - x\right) \left(x + 3\right) \left(2 x + 5\right)}{3 x + 1} \left(x + 4\right) > 0$$
((((4 - x)*(x + 3))*(2*x + 5))/(3*x + 1))*(x + 4) > 0
Detail solution
Given the inequality:
$$\frac{\left(4 - x\right) \left(x + 3\right) \left(2 x + 5\right)}{3 x + 1} \left(x + 4\right) > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{\left(4 - x\right) \left(x + 3\right) \left(2 x + 5\right)}{3 x + 1} \left(x + 4\right) = 0$$
Solve:
$$x_{1} = -4$$
$$x_{2} = -3$$
$$x_{3} = - \frac{5}{2}$$
$$x_{4} = 4$$
$$x_{1} = -4$$
$$x_{2} = -3$$
$$x_{3} = - \frac{5}{2}$$
$$x_{4} = 4$$
This roots
$$x_{1} = -4$$
$$x_{2} = -3$$
$$x_{3} = - \frac{5}{2}$$
$$x_{4} = 4$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$-4 + - \frac{1}{10}$$
=
$$- \frac{41}{10}$$
substitute to the expression
$$\frac{\left(4 - x\right) \left(x + 3\right) \left(2 x + 5\right)}{3 x + 1} \left(x + 4\right) > 0$$
$$\frac{\left(- \frac{41}{10} + 3\right) \left(4 - - \frac{41}{10}\right) \left(\frac{\left(-41\right) 2}{10} + 5\right)}{\frac{\left(-41\right) 3}{10} + 1} \left(- \frac{41}{10} + 4\right) > 0$$
one of the solutions of our inequality is:
$$x < -4$$
Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < -4$$
$$x > -3 \wedge x < - \frac{5}{2}$$
$$x > 4$$
$$\frac{\left(4 - x\right) \left(x + 3\right) \left(2 x + 5\right)}{3 x + 1} \left(x + 4\right) > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{\left(4 - x\right) \left(x + 3\right) \left(2 x + 5\right)}{3 x + 1} \left(x + 4\right) = 0$$
Solve:
$$x_{1} = -4$$
$$x_{2} = -3$$
$$x_{3} = - \frac{5}{2}$$
$$x_{4} = 4$$
$$x_{1} = -4$$
$$x_{2} = -3$$
$$x_{3} = - \frac{5}{2}$$
$$x_{4} = 4$$
This roots
$$x_{1} = -4$$
$$x_{2} = -3$$
$$x_{3} = - \frac{5}{2}$$
$$x_{4} = 4$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$-4 + - \frac{1}{10}$$
=
$$- \frac{41}{10}$$
substitute to the expression
$$\frac{\left(4 - x\right) \left(x + 3\right) \left(2 x + 5\right)}{3 x + 1} \left(x + 4\right) > 0$$
$$\frac{\left(- \frac{41}{10} + 3\right) \left(4 - - \frac{41}{10}\right) \left(\frac{\left(-41\right) 2}{10} + 5\right)}{\frac{\left(-41\right) 3}{10} + 1} \left(- \frac{41}{10} + 4\right) > 0$$
3564 ----- > 0 14125
one of the solutions of our inequality is:
$$x < -4$$
_____ _____ _____
\ / \ /
-------ο-------ο-------ο-------ο-------
x1 x2 x3 x4Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < -4$$
$$x > -3 \wedge x < - \frac{5}{2}$$
$$x > 4$$
Solving inequality on a graph
Rapid solution
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Or(And(-oo < x, x < -4), And(-3 < x, x < -5/2), And(-1/3 < x, x < 4))
$$\left(-\infty < x \wedge x < -4\right) \vee \left(-3 < x \wedge x < - \frac{5}{2}\right) \vee \left(- \frac{1}{3} < x \wedge x < 4\right)$$
((-oo < x)∧(x < -4))∨((-3 < x)∧(x < -5/2))∨((-1/3 < x)∧(x < 4))
Rapid solution 2
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(-oo, -4) U (-3, -5/2) U (-1/3, 4)
$$x\ in\ \left(-\infty, -4\right) \cup \left(-3, - \frac{5}{2}\right) \cup \left(- \frac{1}{3}, 4\right)$$
x in Union(Interval.open(-oo, -4), Interval.open(-3, -5/2), Interval.open(-1/3, 4))