Algebra Review

ALGEBRA REVIEW
LAST SCRIBE POST REVIEW BEFORE THE PRE-TEST!!!

in class, mr.k gave us equations to solve and two word problems.


SOLVING FOR X IN A GIVEN EQUATION


solve for x:

x + 3 = 4 - x

[(2x + 2)/(x^2 + 2x - 15)] + [2/(x - 3)] = 1/(x + 5)

√(w + 5) = √(3 - w) + 1


SOLVING FOR X IN AN ABSOLUTE EQUATION

x + 3 = 4 - x

since we are solving for x in an absolute value equation, x can only equal a positive number because the absolute value represents the distance and distance is always postive. because an absolute value of one side equals the absolute value on the other side, then there is more than one answer meaning that there are 4 equations.

we write down all the possible scenarios and since the absolute value operation means that whatever the number is inside the absolute value brackets, we're only look for its positive. for example, the absolute value of -3 is 3; OR -3 = 3.

mathematics is the science of patterns...

mr. k showed us that theres a pattern when solving for x in an absolute value equation. these are the possible solutions:

first, we remove one expression's absolute value brackets and there we are left with either a positive or a negative of that expression.

here are the possibilitites:

x + 3 = 4 - x;

x + 3 = -(4 - x);

-(x + 3) = 4 - x;

-(x + 3) = -(4 - x);

since we're looking for the positive value, all of these expressions have to be greater than 0.

wow...thats a lot of work just to solve for x but mr. k has lots of neat tricks up his sleeves so he showed us a shortcut.


"x + 3 = -(4 - x)" is the same things as "-(x + 3) = 4 - x" because all whats different is that a -1 was multipled on one side.

"x + 3 = 4 - x" is the same thing as "-(x + 3) = -(4 - x)" because all whats different is that a -1 was multipled on one side.

so really, we only need to solve for x for 2 equations which are:

x + 3 = 4 - x AND x + 3 = -(4 - x)


for x + 3 = 4 - x...

x + 3 = 4 - x
2x + 3 = 4
2x = 1
x = 0.5


for x + 3 = -(4 - x)...

x + 3 = -4 + x
x + 3 = x - 4

notice anything abnormal?

if we subtract both sides by x, then we get 3 = -4 which doesnt make sense.
if we add 4 to both sides we get 7 = 0 or if we subtrace 3 on both sides we get -7 = 0.

THIS IS AN EXTRANEOUS SOLUTION!

so for that equation, x = { }, N/A (not applicable), or "No Solution"

so x can only equal 1/2. BUT! we have to check if 1/2 fits perfectly into the given equation and it makes sense. so we sub in 1/2 for x.

x + 3 = 4 - x
(0.5) + 3 = 4 - (0.5)
3.5 = 3.5

both sides of the equation are equal so x = 3.5


in solving for x in an absolute equation, if the equation was like: x + 3 = 4 - x + 1, we have 4 different equations cuz a -1 isnt being multiplied on both sides. solve it and you'll see what i mean.

its like that because we are not looking for the absolute value of 4 - x + 1. we are solving for x when an absolute value equals another absolute value plus a number.

the next scribe after this lesson was sandy. oh and dont pick me for scribe the next day cuz i was already picked for scribe. its just that my name hasnt yet been crossed off cuz my scribe wasnt yet posted but now it is. so after today's scribe, i think cheriee's next cuz her name hasnt yet been crossed off so she's the next and last scribe. good luck and you'll do a great job like always!!