Linear Inequalities

Linear Inequalities

Definition of Linear Inequalities

• Linear Inequality: An expression involving a linear function that is compared using inequality symbols.

Types of Inequalities

• Greater than ($>$)
• Greater than or equal to ($ extgreater ext{=}$)
• Less than ($<$)
• Less than or equal to ($ extless ext{=}$)

Graphing Linear Inequalities

• Dashed Line: Used for inequalities that do not include equality (i.e., $>$ or $<$).
• Solid Line: Used for inequalities that include equality (i.e., $ extgreater ext{=}$ or $ extless ext{=}$).

Shading Regions

• Shading Above: Indicates solutions for inequalities with $>$ or $ extgreater ext{=}$ (i.e., region above the line).
• Shading Below: Indicates solutions for inequalities with $<$ or $ extless ext{=}$ (i.e., region below the line).

Examples of Solutions to Linear Inequalities

• Solutions to a linear inequality are in the shaded region of the graph.
• Example Inequalities:
  • $y extgreater 2x - 3$
  • $y < -x + 5$

Special Cases

• Example of a special case:
  • $y < 5$ (a horizontal line at $y=5$, shading below the line)
  • $x extgreater ext{= } 2$ (a vertical line at $x=2$, shading to the right)

Graphing in Standard Form

• Standard Form Example:
  • $-4x + 6y > 12$
  • $x - 2y extgreater ext{= } -8$

Testing Points Algebraically

• To determine if a point is a solution of an inequality, substitute the point's coordinates into the inequality.
  • Example
  • Testing the point (0, -4) in the inequality $5x - 2y extless ext{= } 6$ :
    • Substitute: $5(0) - 2(-4) extless ext{= } 6$
    • Result: $8 extless ext{= } 6$ (not a solution)
  • Testing the point (0, -4) in $y > 6x - 1$ returns:
    • Substitute: $-4 > 6(0) - 1$
    • Result: $-4 > -1$ (not a solution)

Testing Points on a Graph

• To verify if certain points are solutions to the graphed inequality, check the coordinates:
  • Points to test:
  • (0, 0)
  • (-3, -4)
  • (-5, 3)

Writing the Linear Inequality

• Given a graph, determine the linear inequality that describes the shaded region.
• Use boundaries and shaded areas to define inequalities clearly, considering whether the line is dashed or solid.