Systems of Inequalities

Example 1

Describe the region $\displaystyle y\ge x+1$.

Draw the solid line $\displaystyle y=x+1$ and shade above it. The boundary is included because of $\displaystyle \ge$.

Example 2

Does $\displaystyle (2,1)$ satisfy $\displaystyle y<x$ and $\displaystyle y\ge0$?

$\displaystyle 1<2$ and $\displaystyle 1\ge0$ are both true, so the point belongs to the system solution.

Problem 4.3.1

Does $\displaystyle (1,3)$ satisfy $\displaystyle y\le2x+1$ and $\displaystyle y>1$?

Problem 4.3.2

For $\displaystyle y<4-x$, is the boundary solid or dashed, and which side is shaded?

Problem 4.3.3

Use the graph to describe the solution of $\displaystyle y\ge x$ and $\displaystyle y\le4-x$. Are $\displaystyle (1,2)$ and $\displaystyle (3,2)$ solutions?

Show worked solution

The region lies on or above $\displaystyle y=x$ and on or below $\displaystyle y=4-x$, including boundaries. $\displaystyle (1,2)$ satisfies both; $\displaystyle (3,2)$ violates $\displaystyle y\ge x$.