Polynomial and Rational Inequalities
Polynomial and rational inequalities mean we are going to learn about both inequalities separately. Polynomials are nothing but the sum of more than one number of monomials. Polynomial inequalities mean it is having a simple difference which is less than and greater than symbol. Likewise rational inequalities mean we have to solve the inequalities which are having the rational form. It will be like `(p(x)) / (Q(x))` . We will see some example problems for solving polynomial and rational inequalities.Example Problems for Polynomial Inequalities:
Solve the polynomial function inequalities 5x2 - 15x + 10 ≤ 0
The given polynomial inequality is 5x2 - 15x + 10 ≤ 0
To solve this inequality first we have to factor the given inequality.
So 5x2 - 15x + 10 ≤ 0
5x2 - 5x - 10x + 10 ≤ 0
5x (x - 1) - 10 (x - 1) ≤ 0
(5x - 10) (x - 1) ≤ 0
5x - 10 ≤ 0 and x - 1 ≤ 0
From this x ≤ 2 and x ≤ 1
So the solution of the inequality lies between 1 and 2.
From the we can learn how to solve the polynomial inequalities. Here we solved the quadratic inequality.Example Problems for Rational Inequalities:
Find the solution of the x from the rational inequalities `(3x + 6) / (2x + 9)` ≥ 1
Given inequality is `(3x + 6) / (2x + 9)` ≥ 1
If we want to solve we have to multiply by 2x + 9 on both sides
So we get,
3x + 6 ≥ 2x + 9
Now we have to add -6 on both sides of the function so we get
3x + 6 - 6 ≥ 2x + 9 - 6
From this 3x ≥ 2x + 3
Add –2x on both sides
3x – 2x ≥ 2x + 3 – 2x
x ≥ 3
So x is always greater than 3.
These are some of the example problems for polynomial and rational inequalities. From the above we can learn how to solve the polynomial and rational inequalities.