Systems of Linear Inequalities in Two Variables

6.2 Systems of Linear Inequalities in Two Variables 
 
System of Linear Inequalities
Definition
A combination of two or more linear inequalities.
Determine The Appropriate Inequality for a Certain Situation
Example of Situation Linear Inequality
\(y\) is greater than \(x\) \(y>x\)
\(y\) is less than \(x\) \(y < x \)
\(y\) is not less than \(x\) \(y\geq x\)
\(y\) is not more than \(x\) \(y\leq x\)
\(y\) is at least \(k\) times \(x\) \(y\geq kx\)
\(y\) is at most \(k\) times \(x\) \(y\leq kx\)
Maximum of \(y\) is \(k\) \(y\leq k\)
Minimum of \(y\) is \(k\) \(y\geq k\)
Sum of \(x\) and \(y\) is greater than \(k\) \(x+y >k \)
Difference between \(y\) and \(x\) is less than \(k\) \(y-x <k\)
\(y\) is more than \(x\) by at least \(k\) \(x-y \geq k \)
 

 

Determine and Shade The Region That Satisfies a System of Linear Inequalities
  1. Mark the region involved for each linear inequality with different and easily spotted markings.
  2. Identify the common region for all the markings inolved.
  3. Shade the common region completely. Make sure that the shading is not outside the common region.
Example
Shade the region

 
Problems Involving System of Linear Inequalities in Two Variables
Example

 

Systems of Linear Inequalities in Two Variables

6.2 Systems of Linear Inequalities in Two Variables 
 
System of Linear Inequalities
Definition
A combination of two or more linear inequalities.
Determine The Appropriate Inequality for a Certain Situation
Example of Situation Linear Inequality
\(y\) is greater than \(x\) \(y>x\)
\(y\) is less than \(x\) \(y < x \)
\(y\) is not less than \(x\) \(y\geq x\)
\(y\) is not more than \(x\) \(y\leq x\)
\(y\) is at least \(k\) times \(x\) \(y\geq kx\)
\(y\) is at most \(k\) times \(x\) \(y\leq kx\)
Maximum of \(y\) is \(k\) \(y\leq k\)
Minimum of \(y\) is \(k\) \(y\geq k\)
Sum of \(x\) and \(y\) is greater than \(k\) \(x+y >k \)
Difference between \(y\) and \(x\) is less than \(k\) \(y-x <k\)
\(y\) is more than \(x\) by at least \(k\) \(x-y \geq k \)
 

 

Determine and Shade The Region That Satisfies a System of Linear Inequalities
  1. Mark the region involved for each linear inequality with different and easily spotted markings.
  2. Identify the common region for all the markings inolved.
  3. Shade the common region completely. Make sure that the shading is not outside the common region.
Example
Shade the region

 
Problems Involving System of Linear Inequalities in Two Variables
Example