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Wize High School Algebra I Textbook (Common Core) > Linear Inequalities in 2 Variables

Systems of Linear Inequatlities

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Systems of Linear Inequalities


When we combine two or more linear inequalities we have a system of linear inequalities. A solution to this system must satisfy all of the inequalities involved.

This means the solution region will be the region that is common to all of the individual solution regions from each inequality.

The solution of the system

y≥−24x+3y<−12\begin{aligned} y &\geq -2 \\ 4x + 3y &< - 12 \end{aligned}
is the overlap of their two regions, seen as the purple triangular region on the left.


Example
Graph the solution region for the system of inequalities

x+y≤4x−2y≥6\begin{aligned} x + y &\leq 4 \\ x - 2y &\geq 6 \end{aligned}

ANSWER:



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Business and Systems of Inequalities



A store plans on selling two different models of digital music players. Player A will cost $250, and player B will cost $300.

When selling model A, it will yield a profit of $25 per player. Model B will yield a profit of $40 per player.

It is estimated that the total demand for both players will not exceed 250 units. This means the maximum number of players sold will not be more than 250.

The store only has $60,100 it can invest for buying these two players.

1. Create a system of inequalities that models this situation.

ANSWER:
Each bit of information allows for an inequality. For example the number of players sold will not be negative, so both A and B are positive numbers. We know the demand will not exceed 250 units. And lastly we know that we only want to invest 60100 when purchasing these two types of players. This gives us the system:

A≥0B≥0A+B≤250225A+260B≤60100\begin{aligned} A &\geq 0 \\ B &\geq 0 \\ A + B &\leq 250 \\ 225A + 260B &\leq 60100 \end{aligned}

2. Graph the solution region of the system?


ANSWER:

3. What does a point in the solution region represent?

ANSWER:
Any point in the solution region will satisfy the given requirements. For example the point (100,100)(100, 100) would be the case of getting 100 of each model. This point does not exceed the demand of 250, and spends less than $60100 that we want to invest. Even though this is a solution it may not be optimal. Other solutions may yield more profit overall.

Find the solution region


Determine the solution region by first finding the solution regions for the individual inequalities. Combine these to find the solution region of the system.

x≥−3x−2y<4\begin{aligned} x &\geq -3 \\ x - 2y &< 4 \end{aligned}

Select the graph the best represents the solution region of the system.

Modeling with a System of Inequalities



On a farm they are looking to mix two types of cattle feed to meet the nutritional requirements of the cows.
Brand X cost $25 per bag and contains
  • 2 units of nutrient A
  • 2 units of nutrient B
  • 2 units of nutrient C
Brand Y cost $20 per bag and contains
  • 1 unit of nutrient A
  • 9 units of nutrient B
  • 3 units of nutrient C
The minimum nutrients needed for the cattle are 12 units of A, 36 units of B, and 24 units of nutrient C.
Select the system of inequalities that best models how the two brands can meet the nutritional information. (Hint, each nutrient will be its own inequality)