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

Mark the region involved for each linear inequality with different and easily spotted markings.
Identify the common region for all the markings inolved.
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

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

Mark the region involved for each linear inequality with different and easily spotted markings.
Identify the common region for all the markings inolved.
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