Question

How do you solve `-3/4x^2+4x-8<0` by graphing?

Answer 1

Graph `y=-3/4x^2+4x-8` with the vertex, y-intercept and (if any) x-intercepts.

Explanation:

Let `y=-3/4x^2+4x-8`.
[Step1] First, you need to complete the square.
`y=-3/4x^2+4x-8`
`y=-3/4(x^2-16/3x)-8`
`y=-3/4{(x-8/3)^2-(8/3)^2}-8`
`y=-3/4(x-8/3)^2+3/4*64/9-8`
`y=-3/4(x-8/3)^2-8/3`

Its vertex is `(8/3,-8/3)` and the graph is convex upward.

[Step2] Determine the intercepts.
Put `x=0` to the function and `y`-intercept is `-8`.
However, there is no `x`-interscepts, since `y` is always
negative.

graph{-3/4(x-8/3)^2-8/3 [-1, 6, -10, 2]}

Therefore, the domain for x that satisfies the given inequation is `-oo< x< oo`, or, any real number.