# How can I graph this inequality on the x-y plane?

I'd like to graph the following inequality on the x-y plane:

$$P(y): (\forall x\in [0,y))\left(\frac{x - \epsilon(1+a)}{a} \leq y \leq \frac{x + \epsilon(1+a)}{a}\right)$$

This is the set of points $(x,y)$ where $\frac{x - \epsilon(1+a)}{a} \leq y \leq \frac{x + \epsilon(1+a)}{a}$ for all $0 \leq x < y$. That is, for a point to be in the set, it must satisfy the inequality, and all points to its "left" with the same y-value must also satisfy the inequality.

$\epsilon \in (0,1]$ and $a \in (0,1]$ are both pre-defined constants.

Ideally, the plot would use multiples or combinations of $a$ or $\epsilon$ rather than numerical axis labels, but if I have to plug in values for $a$ and $\epsilon$ to plot this in Mathematica that would be okay (as an aside, if anyone knows an application that can plot this purely symbolically, please let me know!).

• Please include in your question the code that you already have tried for this purpose. Doing so will increase your chances of obtaining a useful response. – bbgodfrey Mar 6 '16 at 21:58

Manipulate[RegionPlot[
(x - e (1 + a))/a <= y <= (x + e (1 + a))/a && 0 <= x <= y &&
y <= e (1 + a),
{x, 0, 1}, {y, 0, 1}, PlotPoints -> 60],
{{e, .5}, 0.01, 1, Appearance -> "Labeled"},
{{a, .5}, 0.01, 1, Appearance -> "Labeled"}] • Thanks! This is close except I think the constraints are not quite right. For example, as you can see at the top of the plot you posted, there is a shaded region where, for the same $y$, points to its left are not shaded. Those points should be excluded. Also $0 < a \leq 1$ and $0 < e \leq 1$. I think this works: Manipulate[ RegionPlot[(x - e (1 + a))/a <= y <= (x + e (1 + a))/a && 0 <= x <= y && y <= e (1 + a), {x, 0, 1}, {y, 0, 1}, PlotPoints -> 60], {e, 0.01, 1, Appearance -> "Labeled"}, {a, 0.01, 1, Appearance -> "Labeled"}] – sundance Mar 6 '16 at 23:23
• @sundance updated, but i did not see $y <= e (1 + a)$ in your original post. – Vitaliy Kaurov Mar 6 '16 at 23:48
• Right, it wasn't there, it's just a necessary condition to make sure the inequality is satisfied for a given $y$ for all $x$. – sundance Mar 7 '16 at 0:02