Modeling with Systems of Inequalities
Model real-world constraints using systems of inequalities: write each condition as an inequality, graph the system, and identify the feasible region that satisfies all conditions simultaneously.
Key Concepts
Real world problems with multiple constraints, like a budget and an item limit, can be modeled using a system of linear inequalities where variables represent quantities.
You can buy up to 30 tickets ($x + y \leq 30$) and spend up to 1000 dollars ($63x + 20y \leq 1000$). The graph shows all valid combos. For a party, you need at least 15 sodas ($x+y \geq 15$) but can't carry more than 40 pounds ($2x + y \leq 40$).
You're on a shopping spree with two rules: a total item limit and a spending cap. Each rule becomes an inequality! Let $x$ be games and $y$ be snacks. The solution is the shaded zone on the graph showing all possible combinations of games and snacks you can actually afford without getting grounded. It’s your 'safe shopping' zone!
Common Questions
How do you set up a system of inequalities from a word problem?
Identify each constraint in the problem and translate it into an inequality. Define your variables clearly. For example, if x represents hours of work and the problem says 'work at most 40 hours,' write x<=40. Collect all constraints to form the system.
What does the solution region of a system of inequalities represent?
The solution region is the set of all (x,y) pairs that satisfy every inequality in the system simultaneously. In a modeling context, any point within this region is a feasible combination of values that meets all real-world constraints.
How do you determine whether a boundary line is included or excluded from the solution?
If the inequality is strict (< or >), the boundary line is not included and is drawn as a dashed line. If the inequality includes equality (<= or >=), the boundary line is part of the solution and is drawn as a solid line.