Materi Sistem pertidaksamaan Linear dua Variabel (Matematika) - Bagian 1

Summary

The video introduces linear inequalities in two variables and demonstrates how to graph an inequality and identify its solution region.

Key Ideas

• A linear inequality in two variables contains two variables—usually (x) and (y)—each raised to the first power.
• The inequality symbols are:
  • Greater than: (>)
  • Greater than or equal to: (\geq)
  • Less than: (<)
  • Less than or equal to: (\leq)
• In an expression such as (ax + by) compared with a constant, (a) and (b) are coefficients, and the value on the other side is the constant.
• The boundary of the solution region is found by replacing the inequality symbol with an equals sign.

Graphing and Finding the Solution Region

1. Turn the inequality into an equation to find its boundary line.
2. Find the intercepts:
   • Set (y = 0) to find the (x)-intercept.
   • Set (x = 0) to find the (y)-intercept.
3. Plot the intercepts on the Cartesian plane and draw the boundary line through them.
4. Choose the line style:
   • Use a solid line for (\leq) or (\geq), because points on the boundary are included.
   • Use a dotted line for (<) or (>), because points on the boundary are excluded.
5. Test a point on one side of the line—often ((0, 0)), if it is not on the boundary. Substitute its coordinates into the original inequality.
6. Shade the side that satisfies the inequality. The shaded side is the solution region.

Examples Covered

• For the boundary (x + y = 2), the intercepts are ((2, 0)) and ((0, 2)). The point-test reasoning selects the region above the line, corresponding to (x + y \geq 2).
• For the boundary (2x + 3y = 6), the intercepts are ((3, 0)) and ((0, 2)). Testing ((0, 0)) selects the region below the line, corresponding to (2x + 3y \leq 6).

Subtitle Inconsistencies

The inequality signs stated for these examples conflict with parts of their graphing explanations:

• In the first example, the point test and selected region correspond to (\geq), though the example is initially stated as (\leq).
• In the second example, the worked test and selected region correspond to (\leq), though the example is initially stated as (\geq). The test also appears to give the right-hand side as 2; it should be 6.
• One listed “example,” (-3x + 5y = 15), is an equation rather than an inequality as transcribed.

Speakers or Sources

• Presenter: Unnamed presenter from the Math Pi channel.