Algebra: Chapters 9-5 and 9-6 (Friday and Monday), p 417 and 421
Inequalities in 2 variables and Graphing Systems of Linear Inequalities
Inequalities in 2 variables (e.g., `2x – y > 5`) can be solved just like the equation `2x – y = 5`. The only difference is that the solutions lie in a 1/2 plane instead of on the line. The trick to to find what 1/2 plane they satisfy. To solve these equations, we can use a T-chart or solve for the x-intercept (set y=0) and the y-intercept (set x=0) ordered pairs. Plot them on the graph and connect the dots!
Definitions:
- Half-planes: Regions above the line and below the line (or to the left or right), there are 2 half-planes.
- Boundary line: The line in an equation is renamed to this in an inequality.
- If the inequality possesses a < or >, then the line is dashed (or dotted) to show that values ON the line DO NOT satisfy the inequality.
- If the inequality possesses a ≤ or ≥, then the line is SOLID to show that values on the line SATISFY the inequality.
To determine what 1/2 plane to shade, test the ordered pair (0,0) [or (1,1) if that is on the line]. If (0,0) is true for the inequality, then we shade that portion’s [where the (0,0) lies] 1/2 plane as the solution space. If it is NO true, then we shade the opposite 1/2 plane.
If we tackle 2 inequalities, then we do exactly the same for both inequalities BUT we look for where the 2 inequality solutions OVERLAP. The overlap is the solution to both inequalities.
See this link from purplemath.com too!
Here is another for 2 inequalities!!
Math-8, Chapter 7-1, p 330
Problem-Solving Strategy: Work Backwards
“The strategy of working backward can be used to solve problems. To use this strategy, start with the end result and UNDO each step”.