X<5,Y<5 (clear)..but what if X<5, Y-X>10

Question

I'm trying to construct geometric representation of the following:

X<5, Y<5 (that is clear, it will be the area (square) with the corners on the 5s on X and Y axes.

But I am clueless how to proceed if there is relation with the other variable with no precise value, as in:

X<5, Y-X>10. I would ge extremely grateful for graphic representation.

Answer 1

Generally, when plotting a function (of one variable), one puts the values of the function on the $y$ axis, so one does the assignment $y=f(x)$.

Replace $>$ with $=$ in your inequality and find out what is $f(x)$ in your case. That will give you (part of) the outline of your area.

Then look back at the inequality to decide which side of the outline is in your area

Answer 2

To solve a system of linear inequalities:

1. First draw each line. If the inequality is strict ($<$ or $>$), draw a dashed line. If the inequality is weak ($\leq$ or $\geq$), draw a solid line.

2. To determine which region to shade for a given inequality, pick a point that is not on the line corresponding to that inequality. If the inequality is satisfied at that point, shade the side of the line containing the point; if the inequality is not satisfied at that point, shade the side of the line opposite from the point. The region where each inequality is satisfied is the solution set.

For your example \begin{align*} x & < 5\\ y - x & > 10 \end{align*} Since each inequality is strict, draw dashed lines $x = 5$ and $y - x = 10$. Observe that the origin is not on either line. Since $(0, 0)$ satisfies the inequality $x < 5$, shade to the left of the vertical line $x = 5$. Since $(0, 0)$ does not satisfy the inequality $y - x > 10$, shade on the side of the line $y - x = 10$ opposite from the origin. The region that is to the left of the line $x = 5$ and above the line $y - x = 10$ satisfies both inequalities, so it is the solution set. See the diagram below.