Question

How do you graph the compound inequality `3p+6<8-p` and `5p+8>=p+6`?

Answer 1

The first step is to bring both inequalities to a simplest possible form using invariant transformations (that is, those that produce equivalent inequalities).

`3p+6<8-p`
Add `p` to both sides and subtract `6` from both sides.
`4p<2`
Divide both sides by `4`.
`p<1/2` (simplified inequality 1)

`5p+8 >= p+6`
Subtract `p` and subtract `8` from both sides.
`4p>=-2`
Divide both sides by `4`.
`p >= -1/2` (simplified inequality 2)

Now it's easy to combine both simplified inequalities (1) and (2).
The first one restricts `p` from above to be less than `1/2`.
The second one restricts `p` from below to be greater or equal to `-1/2`.
Combining these restrictions, we come to an interval `p` is supposed to be in:
`-1/2 <= p < 1/2`

Graphically, it is represented by an interval on the X-axis `[-1/2,1/2)` where a square bracket on the left indicates that the left border point `p=-1/2` is included into an interval, while the parenthesis on the right indicates that the right border point `p=1/2` is not included into an interval.
Usually, an arrow on the side of strong inequality (`p=1/2`) might indicate that an end point is not included (not on this graph).
graph{sqrt(x+1/2)0+sqrt(1/2-x) 0 [-1, 1, -0.5, 0.5]}