3. Tuesday, July 6

 Some resources if you want to review today's material -- also, try Googling for more videos.
Number of solutions to a linear system
Solving inequalities   Lots more worked examples, some fancy ones
Graphing inequalities on the number line
Graphing linear inequalities
Graphing a system of inequalities

Board shots:

Review of substitution.
When you solve by substitution, there are a couple of steps that need extra notation -- either use arrows to guide the reader visually through the flow, or  label the 2 equations and keep the labels with them through their transformations and use words like "substitute x from A into B"
(note not all the steps are listed on this example -- see the handout)

 
Here is a more complicated example -- it's easier to simplify each equation first to x-term, y-term, and constant term.

Open circles mean the end point is not included.  A coloured circle means that point is included.
To decide which side to colour, you can use a test point -- if it satisfies the inequality statement, that's the correct side.   Also, if the variable is on the left, the inequality sign will mirror the arrowhead of the coloured region.

When solving inequalities, just leave the inequality sign alone unless you divide or multiply each side by the same negative number -- then the sign flips horizontally.  You can avoid this scenario by planning to throw the x-terms to the side that will result in a positive coefficient of x.  Then the last step will be to turn the whole inequality statement around (including the sign) to get the x on the left.

                     

For an inequality with 2 variables, graph the boundary line solid if there is an "...or equal to" sign.  If it's just > ("greater than") or < ("less than"), make the boundary a broken line.  Use a test point to determine which side to colour.
 Otherwise, if the inequality statement starts with "y is greater than....", colour above the line.  "Y is less than..." means colour below the line.







With a system of inequalities, graph each on the same grid and heavily colour the overlapping region.

Here's the Number of Solutions handout filled in.  To find how many solutions a system has, you just have to compare the slopes (different = 1 solution), and if they are the same, look also at the y-intercepts (same slope, same intercept = same line, so infinite number of solutions)  (same slope, different y-intercept means parallel lines, so 0 solutions.)

If you try to solve a system using algebra and the letters disappear, examine the resulting statement.  If it is true (eg 0=0), there is an infinite number of solutions.  If false (eg 0=6), there is no solution.