Question: Solve the inequality \(4t^2 \le 9t-2\)
We rearrange it as follows:
\(4t^2-9t+2 \le 0\)
Then, factor this quadratic equation:
\((4t-1)(t-2) \le 0\)
The critical values (also called endpoints) are: \(t_1=\frac{1}{4},t_2=2\)
So, if we sketched the quadratic \(4t^2-9t+2\), it has x-intercepts (1/4 and 2).
And we want the region below the x-axis (As it is less than 0).
Hence, the region between 1/4 and 2.
Therefore: \(\frac{1}{4} \le t \le 2\)
Here is a sketch: (Using desmos.com)
I hope this helps.
