System of linear inequalities
Graph systems of linear inequalities in Grade 10 algebra by shading solution half-planes for each inequality and identifying the feasible region where all conditions are satisfied.
Key Concepts
A system of linear inequalities is formed by two or more linear inequalities. An ordered pair is a solution of a system if it is a solution of every inequality in the system.
Is $(2, 1)$ a solution for the system $y < 3$ and $x + 2y \leq 4$? Yes! Because $1 < 3$ is true AND $2 + 2(1) \leq 4$ is true. Is $(2, 2)$ a solution for the same system? Nope! While $2 < 3$ is true, $2 + 2(2) \leq 4$ is false. It failed one of the rules!
Think of it as getting into a secret club with multiple rules! To be a member, you must follow every single rule (inequality) perfectly. If you break even one, you're out. The graph's solution is the cool, overlapping shaded area where all the 'rule zones' meet. It’s the spot where every condition is satisfied at once.
Common Questions
How do you graph a system of linear inequalities?
Graph each inequality as a boundary line (dashed if strict </>; solid if ≤/≥), shade the solution half-plane for each, then identify the overlapping shaded region as the solution set.
How do you determine which side of a line to shade?
Test a point not on the line (often (0,0)) in the inequality. If the point satisfies it, shade that side. If not, shade the opposite side.
What does the solution region of a system of inequalities represent?
The solution region (feasible region) is the set of all points (x,y) satisfying all inequalities simultaneously. In optimization problems, the maximum or minimum occurs at a corner of this region.