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For simplicity, multiply both sides of the given inequality by (-1). You will get then an EQUIVALENT inequality
(x-4)*(x+2)*(x-3) < 0 (1)
The polynomial in the left side has simple roots -2, 3 and 4, in ascending order.
Each root is of the multiplicity of 1.
It means that the values of polynomial are of the same sign in each particular interval between the neighbor roots
and change their signs when the variable "x" passes each of the roots.
With it in mind, it is VERY SIMPLE to solve the problem.
In the interval (-oo,-2) the polynomial (1) is negative;
in the interval (-2,3) the polynomial (1) is positive;
in the interval (3,4) the polynomial (1) is negative;
in the interval (4,oo) the polynomial (1) is positive.
So, the solution to the given inequality is this union of two intervals (-oo,-2) U (3,4).
Solved.
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If you want to see other similar solved problems, look into my lessons
- Solving problems on quadratic inequalities
- Solving inequalities for high degree polynomials factored into a product of linear binomials
- Solving inequalities for rational functions with numerator and denominator factored into a product of linear binomials
- Solving inequalities for rational functions with non-zero right side
and
- Domain of a function which is a quadratic polynomial under the square root operator
- Domain of a function which is a high degree polynomial under the square root operator
- Domain of a function which is the square root of a rational function.
in this site.
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Comment from student: Thanks a lot Ikleyn. What is the link of your website? I can share.
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