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Practice Problems to try before the first class, AUG 2014

1. If exp(2*x*) = 3, then what does exp(6*x)* equal?

**Translate**: "9 less than a number is 18"

**Find numbers**such that: A. its square is more than itself; B. its square is less than itself; C. its square is exactly one more than itself; D. its reciprocal is exactly one less than itself; E. its reciprocal and its opposite are exactly the same.

**EXPLORING NEW SOLUTIONS TO OLD PROBLEMS**.

FOR INSTANCE: FIND 2 POINTS ON THE LINE: 2X + 3Y = 18.

SO I PROPOSE, FIND TWO SETS OF NUMBERS WHOSE SUM IS 18. 17 + 1 = 18, AND 9+9 = 18. SO IN THE FIRST CASE, LET'S LOOK AT

17 = 2X, AND 1 = 3Y, SO OUR FIRST SET OF COORDINATES IS (X,Y) = ( 17/2, 1/3). 2nd SET IS VIA, 9=2X, AND 9=3Y, OR X=9/2, & Y=3.

THIS SOLUTION PATH, IS EASIEST FOR GRAPHS OF THE TYPE: X + Y = 9, JUST FIND TWO SETS OF INTEGERS WHOSE SUM IS 9. (6,3) & (4,5), AND STUDENTS CAN GIVE PLENTY MORE IN LESS THAN 30 SECONDS. It is nice to note that for each of the two previous solutions, if we invert the ordered pairs, each "new" ordered pair is also a solution. Note: 3+6=9, so (3,6) AND (6,3) are BOTH solutions, so, if this is true for one more pair, the function is its own inverse, is this one? Hint: invert the second pair (4,5) given above, and test it to: X+Y=9.

THIS WILL ENGAGE ALL STUDENTS INTO WINNING AT MATH. FOR INSTANCE TRY FINDING 4 SOLUTIONS TO: X+Y=11: {(5,6),(6,5),(7,4),(4,7)}, and for X+Y=6: {(2,4),(4,2),(5,1),(1,5)}. so now we have found 3 linear functions, each of which is it's own inverse. Can we generalize to the broader group of linear functions of the form: X+Y=C, where C is a CONSTANT?

THEN MOVE INTO SUBTRACTION; a graph of: X-Y=5. FIND 2 NUMBERS WHOSE DIFFERENCE IS 5, OR, WHEN YOU SUBTRACT THEM, YOU GET 5. (6,1), and (7,2) and (8,3). A pattern occurs, that whenever x is increased by 1, y goes up by one too, this is the slope of the line. m=1. Also notice, this linear function is not it's own inverse, since (1,6) is not a solution to the line, and (6,1) is a solution, to the same equation.

Also, looking at a solution to a(bx+c)+d(fx+g)=h, is found via x=[h-ac-dg]/[ab+df]

Great prep for quadratic equation. So an example could be: Solve 3[2x+1] + 5[3x+7] = 11

(a,b,c,d,f,g,h)=(3,2,1,5,3,7,11). So subbing into our formula, we get: x=[11-3(1) -5(7)]/[3(2)+5(3)] = -27/21 reduce to -9/7 or negative one and two-sevenths.

9%*8% = (9*8)*(%*%)

= 72(%^2) note: (%^2) means "percent squared."

= 0.72%

= 0.0072

final answer.

so PERCENT SQUARED is equivalent to moving the decimal left two places, two times.

Write a problem involving PERCENT CUBED and solve it like this method, it's a good Algebra preview problem.