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Math 101 STUDY GUIDE

Sect/Pg Problems Problem Type
1.1/ 10 55,59,67 Integer Order of Operations
1.2/ 19 41,49,61,69 More Order of Operations
WS#1 1,11,36,46 More Order of Operations
1.3/ 28 27,29,41,45 Order of Operations with Exponents
1.4/ 36 15,21,41,65,69,75 Simple Algebraic Expressions, Evaluating with Substitution, English Phrases Translated
into Algebraic Expressions
WS#2 6,78 Order of Ope rations and Parentheses
2.1A/ 50 11,27,39,49 Solve Linear (1st degree) Equations with Integers
WS#4 6,8,18,19,26,29,44,5
1
Solve Basic Linear Equations (These are problems of types from 2.1A; 2.2A; 2.3A; & 2.4A,
but basically short .. Good practice.)
2.1B/51 51,53,55,57 Word Problems: " Number Type "
2.2A/58 3,9,29,37 Solve 1st Degree Equations with Fractions: Remember you can multiply both sides of the
equation by the LCD to "remove the fractions" from the equation. The safest way is to multiply
each term by the LCD.
2.2B/ 59 41,45,53 Word Problems: "Number Type" with Fractions
2.2C/ 59 43,47,55 Word Problems: "Geometry" with Fractions
2.3A/ 66 1,9,15,23 Solve Decimal Equations : Same idea as the fractional equations ... You can multiply each
term by the correct multiple of 10 to "remove the decimal". Be careful to " add zero 's" when
needed in this process.
2.3B/ 67 29,35,41,43 Word Problems: Percents & Decimals
WS#15 1,3,4,11,12 Word Problems: Percents & Decimals
2.4A&B/ 77 1,7,9,19,25 Symbolic Linear Equations: Solving for One Variable "in Terms of" or by Substitution:
Solve for the unknown cited and treat the other unknowns "like they were constants", adding,
sutracting, multiplying and dividing by them as needed just like you would if they were
numbers. Warning: You have not "solved for the given unknown if it is "in the answer"
(Example: x = 3x + c is NOT solved for x.)
2.4C/ 78 31,45 Symbolic Linear Equations that Require Factoring to SOLVE : Similar to the above
problems but these have more than one term left with the desired unknown once all possible
terms have been combined. Have all terms that have the desired unknown on one side of the
equation and all other terms on the other side of the equation. Then the desired unknown can
be factored out as a common factor on that side. Divide both sides by the quantity left in ( ),
and you will have solved for the desired unknown.
WS#20 2,5,9,44,45 Symbolic Linear Equations that Require Factoring to SOLVE
2.4D/ 78 47,48 Word Problems: "Geometry"
2.4E/78 49, 51 Word Problems: Percents & Decimals: Remember that percents must be changed to
decimals by shifting the decimal place 2 positions to the left. Also, remember that "percents
never stand alone"
: A percent OF something indicates to multiply the decimal equivalent
times that something.
2.4G/78 59, 61 Word Problems: Mixture
WS#16 15,18,20 Word Problems: Mixture
2.5A/ 85 1,7,11,13,17 Linear Inequalities: Notation: Se sure you can use the "new" Number Line notation and
that you can put these into Interval Notation, and vice versa.
2.5B/ 85 31,37,45,59,63 Linear Inequalities: Solving: Remember to keep the inequality symbol facing the same
direction as you go, solving the same way you would solve a linear equation, except: If you
multiply or divide by a negative number, then you must reverse the direction of the
inequality symbol.
Be able to put your answer in any of the 3 forms (Inequality, Number Line,
or Interval Notation.)
3.1/114 Also WS#6) Polynomial Sums & Differences
3.2/120 5,33,49,51,53,63,65 Monomial Products & Quotients: USING EXPONENT RULES
3.3/127 3,9,21,25,37,55,63 Multiplying Polynomials (Includes FOIL)
WS#10 1,9,25,46 (See
also WS#8)
Multiplying Polynomials (Includes FOIL)
WS#7 3,12,24,29,30,47 A mixture of Multiplying and Adding Polynomials
  WARNING** On the Final: Be sure to pay attention to directions as to whether you are to JUST
FACTOR, or SOLVE an equation so that you do NOT confuse these problems from 3.4 -
3.7!
3.4A&B/135 25,39,45,51,55 FACTORING: by Common Factors & by Grouping
3.4C/136 65,71,75 SOLVING EQUATIONS BY FACTORING
3.5A&B/142 5,15,25,39 FACTORING: Difference of Squares (Difference of Cubes may be used for Bonus
3.5C/ 143 57,61,63,67 SOLVING EQUATIONS BY FACTORING
3.6A/151 1,17,25,39,47,53;
See also WS#11
FACTORING: FOIL Type
3.6B/151 57,61,67,73,77,85,9
3
FACTORING: All Methods: Remember to always: 1. Check for Greatest common factor
first,
once that is done... 2. Count the number of terms to help identify the factoring pattern:
Two terms: Difference of squares or difference of cubes (the latter will NOT be on the Final);
Three terms: Foil factor (or trinomial factoring.); Four terms: Possibly by grouping. Always
factor COMPLETELY! See warning** above.
WS#12 1,5,24,40,47 FACTORING: All Methods
3.7A/157 1,17,27,41,53 SOLVING EQUATIONS BY FACTORING: These are equations that have a 2nd degree term
or higher. The idea is to: 1. get your equation set = 0 with terms combined, etc., 2. Then
completely factor that expression. Once it is in factored form, 3. set each factor = 0, since
the only way a product can be 0 is for one or more of the factors to be 0. For most of these
problems, these factors will either be numbers that can not be 0, or will be simple linear
expressions that can be easily solved. See warning** above.
3.7B/157 55,57,59,61 Word Problems: Not assigned.. Possible Bonus Problems..Equation found will be solved
by factoring methods
4.1/171 13,17,21,29,45 Simplifying Rational Expressions (Algebraic Fractions): Remember you can only "cancel"
factors (things that are multiplied both top & bottom. Hint: This requires that there can be only
ONE term top and bottom before cancellation can be considered. Remember proper
exponent rules, and where there are polynomials, get these into completely factored form
in order to simplify.
4.2/177 15,21,25,27,39 Simplifying Rational Expressions: Multiply & Divide: Same idea as above. Remember to
invert to multiply in the case of division before doing any cancellation. Once things are in
factored form and multiplied, you can "cancel" same factors top to bottom whether they are in
the same fraction or in a different one, as long as one is on top and one is on bottom.
4.3/185 9,15,23,31,41,51,
63; See also
WS#21
Simplifying Rational Expressions: Add & Subtract: This is just like adding and subtracting
number fractions, you must get a common denominator and it is very important to get the
Least Common Denominator. Remember in "creating" the LCD, you are essentially
multiplying each fraction by an equivalent of "one" so that the value of your expression
does not change. Warning: Do not confuse this with solving equations that contain
fractions. In Section 2.3, you could multiply by the LCD and remove the fractions, but now
you are Simplifying, and if you multiply by the LCD, you are changing the value of your
expression.. BAD!
5.1A/232 7,25,35,37,39 Integers as Exponents with Numbers: Exponent rules are the same even though we are
now considering negative integer exponents also. The negative exponent just means to take
the inverse of whatever is raised to that exponent. If everything is multiplied and/ or divided,
this essentially means that you can "move" a factor with its exponent "down" or "up" in a
fraction by changing the sign of its exponent. Refer to class notes for rules and examples.
Warning: A negative exponent does NOT make the number negative. (Example 3 "to the -1
power" does NOT equal -3... It is equal to 1/3. Be sure to "evaluate" all numbers raised to
exponents... That is, get a number equivalent. (Example, don't leave 5 raised to the 3rd
power... This should be evaluated as 125.)
5.1B/232 43,53,57,59,71 Integers as Exponents with Algebraic Expressions: Same principles as above, except that
you will not be able to evaluate a variable raised to a power. NEVER leave a negative
exponent
in an answer.
5.1C/232 75,81,83 Integers as Exponents with Algebraic Ex pressions with 2 Terms : When more than one
term is involved, then you are not in factored form. First, change each factor to the correct
position with a positive exponent
instead of a negative one. Now, it will probably "look
like" an addition of fractions
problem like those in 4.3. Proceed as with those problems.
WS#18 3,4,5,6,7,11,14,16,1
9,23,24,28,36,39,43
Basic Exponent Problems
5.2A/243 1,3,5,7,15 Basic Radicals with Numbers: These "come out of the radical exactly."
WS#3 2,6,8,11,19,20,39,53 Basic Radicals with Numbers
5.2B/243 23,31,35,41 Simplifying Radicals with Numbers: Here, "split the number into factors" so that one is a
perfect square (hopefully the largest one possible.) and the other is not. Take the square root
of the perfect square and multiply it on the "outside" and leave the other number "inside." If
you did not find the largest perfect square the first time, you will have to repeat this process.
WS#13 2,8,32,41,43 Simplifying Radicals with Numbers
5.2C/244 45,51,61,69,73 Simplifying Radicals: Rationalizing Denominators with Square & Cube Roots with
Numbers
Only.
To "rationalize the denominator" means to "remove the radical(s) from the
denominator". This must be done in such a way that the "value of the expression" is not
changed.
Same concept of not changing value as we faced with fraction expressions, so
whatever we do here to change the form will be done by multiplying by an equivalent of
"one"
so that value is not changed. The idea in both the square root and cube root case is to
figure out what to multiply the denominator by so as to "create" a perfect square or a
perfect cube
, respectively, under the radicals, so that the radical can then be simplified
"exactly, " thus "removing" the radical. Remember that you must multiply square root by
square root and cube root by cube root, in order to be able to multiply the "insides" as desired;
and you must multiply the "top" by the same thing as the "bottom", so that you have
merely multiplied by "one."
5.3A/249 1,7,9,11 Simplifying Radicals: Sums with Numbers Only: You will just be simplifying radicals as in
5.2B, but you will find that the results are just like having "like terms", like adding 2x + 5x =
7x, except the "x" is a square root or a cube root.
WS#5 3,8,9,15,17,18,19,23
,30,31,33,51
Exact Radical Problems: Practice with different combinations of operations with roots, but all
the answers are "no radicals." Evaluating "exact radicals" with variables involved: Here
we are taking square or cube roots of a variable, say x, raised to a power that will come out of
the radical exactly. The square root of x raised to the n power can be written as x raised to the
n/2 power; thus if n is an even number, the expression "comes out" of the radical exactly.
Similarly, the cube root of x raised to the n power is x raised to the n/3 power, so in the case
that n is a multiple of 3, this expression will "come out" of a cube root exactly.
5.3B/249 27,29,37,55 Simplifying Radicals: Algebraic, & Rationalizing Denominators with Algebraic Square &
Cube Roots:
As in the above discussion: When we are faced with the situation of n not
being the correct multiple to come out of the radical exactly, we pick the largest integer less
than n that is the correct multiple, then use the exponent rule to express x raised to the n
power as x raised to this next integer times x raised to whatever is necessary to "get back up"
to "x to the n" (Example cube root of x raised to the 14th power: The largest integer less than
14 that is divisible by 3 is 12. Thus x to the 14th = x to the 12th times x to the 2nd. Cube root
of x to the twelth is excatly x to the 12/3 or x to the 4th, and the x to the 2nd stays under the
cube root. Same idea as with numbers is used for rationalizing.
5.7/273 3,9,13,21,29 Scientific Notation
7.2/357
7.5B/387
7,17,25,29,31xxxxxx
xx49 (from 7.5)
Graphs of Lines: You will have to identify both x- & y-intercepts. You need to find these points
algebraically .... Not estimating through the graph. You will also have to find the slope of the
line. You will have to graph 3 points, including the x- & y-intercept. May use the slope to find
the 3rd point. or "an x,y chart". Identify equation & slope of horizontal &vertical lines
WS#14 1,5,7,8 Uses problems in 7.2 to do the following: Putting equations of lines into Slope-Intercept
form;
finding the slope; finding the slope of any perpendicular line; finding x- and y-intercepts.
Also be able to identify intercepts from a graph as we did on PT#6, problem
#25.
7.3/364 1,5,13 Graphs of Linear Inequalities in Two Variables (Shade Graphs): (Directions will be the
same as on Major Test #3 ... Plot x- & y-intercept (a 3rd point if you wish to be safer!) for the
line and then show your "Test Point" on the graph and the work for your "Test" in your problem,
and shade properly.)
7.4/374 1,5,9 Distance Between Two Points: In words: to find the distance between two points, do the
following: Take the difference of the "x-values" and then square, do the same thing for the
difference in the "y-values" then square. Add these two numbers together, then take the
square root of the result. The formula is on page 366.
7.4/374 19,25,27 Slope Between Two Points: In words, this is Rise over Run. Given two points, take the
difference in the "y-values" and divide by the difference in the "x-values", being careful to keep
the same order (2nd minus 1st) on both top and bottom. The formula is on page 369. Note:
Also be able to identify slope of a line by looking at its graph as we did on PT #6, problem
#26
7.5/386 3,11,19,35,39 Writing Equations of Straight Lines: (Be able to put into both Standard Form & Slope -
Intercept Form, depending on which is requested. Use our WS#22 Answers to 7.5 in y=mx+b
form to practice that.
PT#6 #16 - #19 Be able to match the graph of a horizontal or vertical line to its equation and to identify
its slope as on PT #6.
11.1/602 1,3,13,21,23,33,3 Systems of Linear Equations (You will need to know BOTH the Substitution and the
Elimination by Addition Methods) Remember to give your answer as an ordered pair with the x-value
first and the y-value 2nd.
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