How do you manipulate inequality in quadratics or manipulating inequality signs in general (Grade 10)??

Question

I have been assigned to do a quadratic word problem. One of them included solving the quadratic inequation of $-5t^2 + 20t > 15$. I am not sure how to start off and I have no memory of doing inequalities. Thank you for trying!!

Answer 1

You can write your inequality after dividing by $-5$ and we get $$t^2-4t+3<0$$ This is equivalent to $$(t-3)(t-1)<0$$ Can you solve this? Write $$-5t^2+20t-15>0$$ then we divide by $-5$ we get

$$t^2-4t+3<0$$ solving the equation

$$t^2-4t+3=0$$ we get $$t_1=3,t_2=1$$

Answer 2

Consider $f(t)=5t^2-20t+15$. You know about parabolas, hence you know that $f$ takes negative values exactly between its zeroes.

Answer 3

Solving an inequality is much like solving an equality - but you must remember to change the sign if you multiply or divide by a negative number. In general, the operations used to solve equalities and inequalities (addition, subtraction, multiplication, division, squaring both sides, etc.) can be viewed as applying a function to both sides. If the inequality holds, and the function that you are applying is increasing, then it still holds after applying the function. If the function is decreasing, then the opposite inequality is now true (change the sign). If the function is neither increasing or decreasing, then it gets more complicated and you must evaluate cases (think squaring $x < y$ when $x$ is large negative and $y$ is small positive).