Examples of solving systems of inequalities by graphing

📚 Quick Study Guide

• 🔍Definition: A system of inequalities is a set of two or more inequalities with the same variables.
• 📈Graphing Linear Inequalities:
  • 🖊️ Replace the inequality sign with an equals sign and graph the boundary line.
  • If the inequality is strict ($<$ or $>$), draw a dashed line. If it is non-strict ($\leq$ or $\geq$), draw a solid line.
  • 🧪 Choose a test point not on the line and substitute its coordinates into the inequality. If the inequality is true, shade the region containing the test point; otherwise, shade the other region.
• 🧩Solving a System of Inequalities by Graphing:
  • ✏️ Graph each inequality in the system on the same coordinate plane.
  • 🎨 The solution set is the region where the shaded regions of all the inequalities overlap. This overlapping region is sometimes called the feasible region.
  • 📌 The vertices of the feasible region are the points where the boundary lines intersect. These points can be found by solving the system of equations formed by the boundary lines.

Practice Quiz

1. Which point is a solution to the system of inequalities: $y > x + 1$ and $y < -x + 5$?
   1. (0, 0)
   2. (2, 4)
   3. (1, 3)
   4. (3, 1)
2. Which inequality is represented by a dashed line on a graph?
   1. $y \geq 2x - 1$
   2. $y \leq -x + 3$
   3. $y > x - 4$
   4. $y \leq 5$
3. What does the overlapping shaded region in a system of inequalities represent?
   1. No solution
   2. The set of all solutions to one inequality
   3. The set of all solutions to the system of inequalities
   4. The boundary lines
4. Which system of inequalities represents the shaded region above the line $y = x$ and below the line $y = -x + 4$?
   1. $y < x$ and $y > -x + 4$
   2. $y > x$ and $y < -x + 4$
   3. $y > x$ and $y > -x + 4$
   4. $y < x$ and $y < -x + 4$
5. Which of the following is a vertex of the feasible region defined by the inequalities $x \geq 0$, $y \geq 0$, $x + y \leq 5$?
   1. (1, 1)
   2. (2, 2)
   3. (5, 0)
   4. (1, 5)
6. What type of line should be used to graph the inequality $y \leq 3x + 2$?
   1. Dashed line
   2. Solid line
   3. Dotted line
   4. No line
7. If the solution region of a system of inequalities is empty, what does this indicate?
   1. The system has infinitely many solutions.
   2. There is no solution to the system.
   3. The system has only one solution.
   4. The system has two solutions.