Pre-Algebra: Chapter 8-9
Graphing Inequalities
Given an inequality, treat it as an equality and using the x and y intercepts, find the solution to the equality. Plot it on your graph paper. If the inequality is just <, or >, then the boundary line (the line you draw connecting the dots) will itself be dotted or dashed. This mean that the points on the line are NOT part of the solution. If the inequality has a ≤ or ≥, then the line will be solid, signifying that the line is part of the solution.
There are 2 ½ planes on the graph, one side of the boundary line that belongs to the solution set (this side will be shaded as part of the solution) and the other side of the line that does not satisfy the inequality.
Now to figure out what ½ plane to shade, pick a point [I like to pick (0, 0) or (1, 1)] and try those (x, y) values in the inequality. IF the point chosen makes the inequality TRUE, then shade that part of the plane. IF the point chosen does not satisfy the inequality, then shade the OPPOSITE side ½ plane.
Here are some more examples!
Algebra: Chapter 6-6
Factoring by Grouping
The distributive property can be used to factor some polynomials with 4 terms. Remember a(b + c) = ab + ac and its reverse, ab + ac = a(b + c)
Consider the cubic, x^3 + x^2 +2x + 2
There is NO common factor to ALL terms. We can however, factor x^3 + x^2 and 2x + 2 separately:
(x^3 + x^2) = x^2(x + 1) and
2x + 2 = 2(x + 1).
Therefore, using ab + ac = a(b + c), we have a = (x + 1) and b = x^2 and c = 2. The final factoring is then (x + 1)(x^2 + 2)!
A “Factoring in Pairs” discussion follows here, it is pretty good, just a different name!