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solve the Polynomial inequality and express the solution in set notation.
(3p-2p^2)/(4-p^2)<(3+p)/(2-p)
You want
(3p-2p^2)/(4-p^2) - (3+p)/(2-p) < 0
p(3-2p)/(2-p)(2+p) - (3+p)(2+p)/(2-p)(2+p) < 0
[p(3-2p) - (3+p)(2+p)]/(2-p)(2+p) < 0
(3p - 2p^2 - p^2 - 5p - 6)/(2-p)(2+p) < 0
(3p^2 + 2p + 6)/(2-p)(2+p) > 0
The numerator is always positive.
So, we want the region where 4-p^2 is positive.
That is, -2 < p < 2
Just to check, graph the two functions, and you'll see that this is the case.
Factor the polynomials by pulling out the GCF
6r^2+12r-15
Sorry, I posted in the wrong place :(
To solve the polynomial inequality and express the solution in set notation, follow these steps:
Step 1: Simplify the inequality equation as much as possible.
Start by multiplying both sides of the inequality by the denominator of the fractions to eliminate them. This will help in simplifying the inequality.
(3p - 2p^2)(2 - p) < (3 + p)(4 - p^2)
Simplify the equation:
(6p - 3p^2 - 4p + 2p^2) < (12 - 4p - 3p^2 - p^3)
Combine like terms:
(6p - 2p + 2p^2 - 3p^2 + 3p^2 - p^3 - 12) < 0
Simplify further:
(p^3 - 4p^2 + 4p - 12) < 0
Step 2: Factorize the polynomial equation if possible.
To find the roots of the polynomial equation, you can try factoring it. However, in this case, the polynomial does not appear to be factorable using rational numbers or simple factoring methods. Therefore, we will use alternative methods to solve the inequality.
Step 3: Use a graphing calculator or software.
One way to determine the solution to the inequality is by using a graphing calculator or software. Plot the graph of the polynomial equation y = (p^3 - 4p^2 + 4p - 12) and find the regions where the graph is below the x-axis.
The solution to the inequality is the set of x-values where the graph lies below the x-axis.
Step 4: Express the solution in set notation.
The solution to the inequality is the set of x-values where the graph lies below the x-axis. We can express this using set notation { p | (p^3 - 4p^2 + 4p - 12) < 0 }. This represents the set of values for which the inequality is true.