Explanation:
To solve this inequality:
4y^2 - (5 - 2y)(1 + y) < (2y + 1)(3y - 2) - 1
you can start by simplifying both sides of the inequality.
First, you can use the distributive property to expand the right side of the inequality:
(2y + 1)(3y - 2) - 1 = 6y^2 - y - 2
So the inequality becomes:
4y^2 - (5 - 2y)(1 + y) < 6y^2 - y - 2
Next, you can distribute the negative sign to simplify the left side of the inequality:
4y^2 - (5 - 2y)(1 + y) = 4y^2 - 5y + 2y^2 + 5y - 2y^2 = 2y^2 + 4y
So the inequality becomes:
2y^2 + 4y < 6y^2 - y - 2
Now, you can simplify both sides of the inequality by combining like terms:
4y + 2 < 4y^2 - y
Then, you can move all terms to one side of the inequality:
4y^2 - 5y - 2 > 0
Finally, you can solve for y by factoring the quadratic equation:
4y^2 - 5y - 2 = (4y + 1)(y - 2) > 0
The solution to the inequality is the set of all values of y that make the expression greater than zero. Since the product of two factors is greater than zero if and only if both factors are positive or both factors are negative, the solution to the inequality is:
y < -1/4 or y > 2
So the solution to the original inequality is:
-1/4 < y < 2
I hope this helps! Let me know if you have any further questions.