Question

How do you solve `6x^2+x>1`?

Answer 1

`color(green)(x < -1/2)` or `color(green)(x > 1/3)`

Explanation:

`6x^2+x > 1` is equivalent to `6x^2+x-1 > 0`

Factoring `6x^2+x-1`
we have `(3x-1)(2x+1)`
which will be equal to zero when `x=1/3` or `x=-1/2`

We not interested in when `6x^2+x-1` is equal to zero.
Rather we are interested in the ranges set up by these "boundaries"

That is we are interested in the ranges:
`color(white)("XXX")x < -1/2`
`color(white)("XXX")x in (-1/2,1/3)`
`color(white)("XXX")x > 1/3`

We can test each of these ranges by picking any easily evaluated sample value of `x` within each range
and then seeing if that value satisfies the required inequality.

For example:
`color(white)("XXX")`in the range `x < -1/2` we could use `x=-1`
`color(white)("XXX")`in the range `x in (-1/2,1/3)` we could use `x=0`
`color(white)("XXX")`in the range `x > 1/3` we could use `x=+1`

#{:(,"|","in range " x < -1/2,"in range "x in (-1/2,1/3),"in range "x > 1/3), (,"|",underline("sample: " x=-1),underline("sample: "x=0),underline("sample: "x=+1)), (6x^2+x,"|",color(white)("XXX")5,color(white)("XXX")-1,color(white)("XXX")7), (6x^2+x > 1?,"|","True","False","True") :}#