Download Solving Mania KCAS: 7.EE.4, 8.EE.7, A.SSE.1, A.CED.4, A.REI.1, A

Solving Mania
KCAS: 7.EE.4, 8.EE.7, A.SSE.1, A.CED.4, A.REI.1, A.REI.3
Operations with Integers
7.NS.1
The mathematical operations are addition, subtraction, multiplication, division, powers, and roots.
INVERSE OPERATIONS undo each other. Addition & Subtraction are Inv Ops. Mult and Div are inv ops.
Square roots and squares are inv ops. Cube roots and cubes are inv ops. Fourth roots and fourths are inv ops, etc. A
FRACTION BAR indicates the operation of division. Can you simplify COMPLEX FRACTIONS? You cross-multiply
only in a proportion (equation of ratios; 2 fractions = to each other). You divide out common factors (some folks call
CROSS-CANCEL) only when you MULTIPLY fractions. Define RELATIVELY PRIME.
Opposites or ADDITIVE INVERSES are two numbers that sum to zero like -¾ and +¾ . Zero is its own opp.
Reciprocals or MULTIPLICATIVE INVERSES are two numbers whose product is 1 like -¾ and -4/3. Zero has
no reciprocal. The number one is its own reciprocal.
Absolute Value is the distance from the ORIGIN on a number line. Absolute Value is NEVER NEGATIVE.
Integers are the whole numbers and their additive inverses (opposites). 0, ±1, ±2, ±3, …
ADDITION
pos + pos = pos
neg + neg = neg
Pos + neg = ???
To add numbers with
different
signs,
THINK of it like subt
of abs values & take
the sign of number
with greater abs value.
SUBTRACTION
MULTIPLICATION
pos * pos = pos
neg * neg = pos
pos * neg = neg
neg * pos = neg
DIVISION
pos ÷ pos = pos
neg ÷ neg = pos
pos ÷ neg = neg
neg ÷ pos = neg
Use the definition of
An EVEN number of DIVISION BY ZERO
subtraction, and
THEN follow addition negative factors yields IS UNDEFINED!!!
a POSITIVE product;
rules
an odd number of
negative factors yields
a negative product.
Factor * Factor = Product. Factors are separated by multiplication symbols.
Do integer quizzes using rules above, pictures, or common sense
An equation is a mathematical sentence that contains an = (equal sign). It can be solved or graphed.
There are absolute value equations like |x| = 7.
There are literal equations that contain 2 or more variables (like formulas, y=2x+3, etc.).
There are linear equations that graph as lines and are first degree (degree is exponent on variable). y=x
There are quadratic equations that graph as parabolas (U-like smooth curves) and are second degree. y=x2
There are cubic equations that graph as “S-like” smooth curves and are third degree. y=x3
An inequality is a mathematical sentence that contains a < ≤ > ≥ ≠ < ≤ > ≥ (inequality symbol). It can
be solved or graphed.
< is read in words as “IS less than.”
≥ is read in words as “IS greater than or equal to.”
≠ is read in words as “IS NOT equal to.”
In math, “less than” means subtract and “greater than” means add.
2 less than 3 means 3-2. 2 is less than 3 means 2 < 3.
Terms: Separated by addition or subtraction signs NOT IN PARENTHESES.
Like Terms: Sep by + or – signs not in ( ) AND their variables and the exponents on those variables match
exactly. You can add and subtract like terms only. You can multiply anything.
Equations NOTES & Home Learning Practice: Page 1 of 8
3x2 + 4x2 = 7x2
3x2 * 4x2 = 12x4
3x2 + 4x3 cannot add because the two terms are NOT alike
3x2 * 4x3 = 12x5
Coefficient: a number that is multiplied by a variable. It is typically written in front of the variable. If no
coefficient is present, it is understood to be one. Therefore, x and 1x are equivalent.
To graph inequalities, it is best to write variable first. If 3 < x, then x > 3. If varible first, shade the way it points. Solid
circle is equal is part of ineq; OPEN CIRCLE if just < or > with no equal part.
3x6-4x3-10 is a TRINOMIAL in descending order of exponents. It has 3 terms, is 6th degree, has a leading
coefficient of 3 and a CONSTANT TERM of -10.
PROPERTIES TO RE-WRITE EQUATIONS/Ineq AND PRESERVE EQUALITY/ineq.
The addition property of equality states that you can ADD the same quantity to both SIDES of an equation & preserve the
equality. If a = b, then a+c = b+c
The MULTIPLICATION property of equality states that you can MULTIPLY every TERM of an equation by the same
quantity and preserve the equality. If a = b, then ac = bc
There is also a subt prop of eq, div prop of eq, addn prop of inequality, subt prop of inequality, etc. What do you think
they say?
PROPERTIES TO RE-WRITE EXPRESSIONS
Additive Identity Property - you can add zero to an expression and not change it. a+0=a
Multiplicative Identity Property-you can multiply a term by 1 and not change it. a*1=a
1.
2.
3.
4.
5.
6.
7.
8.
8 steps to solve linear equations OR inequalities
*******GOAL: isolate the variable for which you are solving!!!*******
 * Means step is OPTIONAL.
* Use the definition of subtraction to change – to + - of the subtrahend.
* Clear fractions by multiplying EACH TERM on both sides of the eqn/ineq by the LCD of ALL
fractions. Clear decimals by multiplying EACH TERM on both sides of eqn/ineq by the proper power of
10 that would eliminate ALL decimals.
Use the distributive property to clear parentheses.
Combine Like Terms on the LEFT of the = or ineq. Combine Like Terms on the RIGHT side of the = or
ineq.
Get all terms containing the variable on only one side of the = or ineq by using the additive inverse
property. Make sure to add the same thing to BOTH sides of the = or ineq.
Use the additive inverse property to isolate the variable term (get rid of the constant term). Make sure to
add the same thing to BOTH sides of the eqn or ineq. (add or subt to SIDE to make ZERO; diff signs).
You move terms by addn or subt. Remember to only add LIKE TERMS and write it exactly as it is
UNDER it's like term.
Use the multiplicative inverse property to isolate the variable. Make sure to multiply BOTH sides of the
eqn or ineq by the same thing. (Mult or Div to make 1; same sign.) ***REMEMBER that if you
MULTIPLY or divide both sides of an INEQUALITY by a NEGATIVE number that you must
reverse the direction of the inequality symbol!!!!!!!!! Every TERM must be mult/div by same #.
Remember that you move factors by mult or div.
Simplify and CHECK YOUR SOLUTION. Substitute your solution into the original eqn/ineq and see if
it makes a true eqn/ineq.
When solving linear equations, you can get the following “types” of results (3 possibilities/cases for solutions):
I.
II.
X=4. This means you have one solution, and it is 4.
-4 = 4 or 2=7 or 0=6, etc. If you get something untrue like this, there are NO SOLUTIONS to your
equation.
Equations NOTES & Home Learning Practice: Page 2 of 8
III.
4 = 4 or 2=2 or -6=-6, etc. If you get something always true like this, there are INFINITELY MANY
solutions to your equation, and they are ALL REAL NUMBERS. That means any real number when
substituted in for the variable will produce a true equation.
WHEN SOLVING ABSOLUTE VALUE EQUATIONS (AVE), YOUR GOAL IS TO FIRST ISOLATE
THE ABS VALUE AND THEN MAKE AN “OR” STATEMENT. When you make that statement, you rewrite what you see without bars, then write “OR,” then re-write what you see again except with the OPPOSITE
of the last number. Lastly, solve the TWO equations. There are 3 cases (or possibilities) when solving AVE.
If |x|=7, then x=7 OR x = -7.
If |x|=0, then x=0
If |x| = -3, then there are NO SOLUTIONS because abs val can’t yield a neg.
When solving abs value inequalities (AVI), isolate the abs value FIRST, and then use the acronym GOLA.
***Iso abs value, & if positive number (like first two cases below) then copy what you see w/o bars, write word
"or" OR "and" based on GOLA, then copy again BUT FLIP SIGN & make # NEGATIVE. If neg # or zero
after isolating abs value, USE COMMON SENSE from last 6 cases below.
There are 8 possibilities/cases for AVI. You will solve a COMPOUND INEQUALITY.
If |x| < 7, then x < 7 AND x > -7. If written using Set Notation: {x | -7 < x < 7}
If |x| > 7, then x > 7 OR x < -7.
If |x| < 0, then NO SOLUTIONS because #s less than 0 are negative, and AV can't be neg.
If |x| < 0, then x = 0 because #s less than 0 are negative, and AV can't be neg.
If |x| > 0, then ALL REAL NUMBERS are solutions because AV will always be 0 or positive.
If |x| > 0, then ALL REAL NUMBERS EXCEPT 0 are solutions because AV will always be pos
except when it's 0.
If |x| < -3, then there are NO SOLUTIONS because abs val can’t yield less than a neg.
If |x| > -3, then ALL REAL NUMBERS are solutions because AV will always be gr than a neg.
When graphing COMPOUND INEQUALITIES, "and" graphs usually go in and "or" graphs usually go out.
How do you know if it's an open circle or closed circle?
When solving literal equations, isolate the variable for which you're solving by following the 8 steps above.
Remember that - ½ = (-1)/2 = 1/(-2) It is easier to work with fractions when neg is in numerator.
Remember that x/2 = (½) x and (2/3) x = (2x)/3
ONLY USE HORIZONTAL FRACTION BARS and keep items in numerator if allowed.
If ALL things on one side of = sum to zero, remember to keep a zero there.
Follow order of ops and watch neg signs.
Why aren't 2x and x2 equivalent? Why aren't x cubed and x tripled equivalent?
Word Problems:
First, state your HEADING. Then write it like you read it unless you see "less than."
Tips: Let variable be item listed last. Make heading for each unknown. Use info to make equation and solve it.
Then substitute the value of your variable in the heading and ANSWER THE QUESTION ASKED IN A
COMPLETE SENTENCE WITH UNITS.
Remember that “is” means “=” and that “of” means "multiply."
To change a DECIMAL to a PERCENT (1. = 100.%), move decimal point 2 PLACES RIGHT.
To change a PERCENT to a DECIMAL (100.% = 1.), move the decimal point TWO PLACE LEFT.
Words like sum, diff, etc. indicate a GROUP and must have ( ) around the entire sum/diff/etc.
PERCENT OF CHANGE = |original # - new # | / (ORIGINAL NUMBER)
If original to new goes up, it's a percent of increase. If orig to new goes down, it's a percent of decrease.
If angles or sides are CONGRUENT, their measures are EQUAL. Equivalent means = too.
<A is read as angle A. m<2=300 is read the measure of angle 2 is equal to 30 degrees.
If 2 angles are COMPLEMENTARY, their measures sum to 90. (If picture, 2 adjacent angles that form a RIGHT ANGLE.) Look for "square" or stmt rays perpendicular.
If 2 angles are SUPPLEMENTARY, their measures sum to 180. (If picture, 2 adjacent angles that form a STRAIGHT ANGLE.)
REGULAR means all SIDES are congruent to each other AND all ANGLES are congruent to each other.
In a polygon, the sum of the interior angle measures is given by the expression (n-2)180 where n is the # of sides the polygon has.
In a triangle, the int < measures sum to (3-2)180 or 180. In a quadrilateral, the int < meas sum to (4-2)180 or 360, etc.
Equations NOTES & Home Learning Practice: Page 3 of 8
Why will the heading for CONSECUTIVE INTEGER word problems always be x, x+1, x+2, x+3, x+4, etc. (as many as needed, example on left is 5 consec integers)?
Why will the heading for CONSECUTIVE EVEN INTEGER word probs always be x, x+2, x+4, x+6, x+8, etc. (as many as needed, example on left is 5 consec EVEN ints)?
Why will the heading for CONSECUTIVE ODD INTEGER word probs always be x, x+2, x+4, x+6, x+8, etc. (as many as needed, example on left is 5 consec ODD ints)?
In word problems, write it like you read it unless you see the word “LESS THAN.”
ADDITION
Sum (use parentheses)
Plus or add
Added to
SUBTRACTION
Difference use ()
minus
subtract
Subtracted from
Decreased by
MULTIPLICATION
Product use ()
multiply
times
Twice (2 times)
Tripled (3times)
DIVISION
Quotient use ()
Divided by or divide
Greater than,More than
**Less Than,Fewer than OF
***LESS THAN means reverse order of how you say/read & subtract
“IS LESS THAN” means the inequality < ; like two IS less than 3 would be written 2 < 3 .
2 less than 3 means 3-2 .
“SQUARED” means raise to the SECOND POWER; 7 squared is 72.
“CUBED” means raise to the THIRD POWER; 4 cubes is 43.
The SQUARE ROOT means take a root; the square root of 100 is √100 .
The CUBE ROOT means take a cube root; the cube root of 27 is 3√27 .
“IS”, equals, and equivalent means = and “OF” means multiply.
CLASSROOM PRACTICE
Solve the following equations
LINEAR (1st degree): Please solve the following equations.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
2 + 3 (x+4) = 20 ***No 5 in this problem; watch order of operations
1 - 4(x-2) = -3
*** there should be a +8 in this problem; watch negative signs
2x + 3(x+4) = 2x + 4 + 1 *** Combine LIKE TERMS on one side of equation; then get x-terms on same side
2(x+3) = 2x - 6 ***NO SOLUTIONS
7(x-4) = 7x - 28 ***ALL real numbers are solutions
2x + 4x - 7 + 3 - 4 (x-2) = x + 4 ***No -10 in this problem; watch sign in FRONT of TERM; x=0 is solution
¾ = x/5 *** In a proportion (fraction = fraction), CROSS PRODUCTS ARE EQUAL!!!!!!!!!!!!!!!!
x/2 = 7/8
1 + ¾ = x/5
(x+3)/4 = 7/2
(x+5)/-3 = (x-7)/2
(x+2)/-3 + 5 = (x-1)/2 - 7
½(x+2) + 3x = x - ¾
1.35(x+2) = x + 2.7
Quadratic, cubic, quartic, pentic, etc. (Negative) even = Positive
1.
2.
3.
4.
5.
6.
7.
(Negative) odd = Negative
2
X = 64
X2 = 61
X2 = -4
X3 = 8
X3 = -8
X10 = -1
X10 = 1
Equations NOTES & Home Learning Practice: Page 4 of 8
Absolute Value: Please solve the following equations.
1.
2.
3.
4.
5.
6.
7.
|x| = 12
|x| = -4
|2x + 4| = 16
|2x + 4| = 0
|2x + 4| = -8
2 - |x + 7| = -3 ***ISOLATE ABSOLUTE VALUE FIRST, then make or statement.
3+2 |3x + 1| - 4 = 5
Literal: Please solve the following equations for the indicated variable.
1.
2.
3.
4.
Solve F=ma for a
P = 2 l + 2 w for w
2a + 3b - dfg = c for b
2a + 3b - dfg = c for f
Solve the following inequalities & GRAPH the solution set
Linear:
1.
2.
3.
A > -4
B<6
3a + 6 < 7
Absolute Value:
1.
2.
3.
4.
5.
6.
7.
8.
9.
|2x + 4| < 0
|2x + 4| < 0
|3x - 1| > 0
|3x - 1| > 0
|3x - 1| > -4
|2x + 4| < -8
|2x| > 14
|2x + 3| < 11
2 |x| - 4 < -2
Solve the following WORD PROBLEMS:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
What percent of 5 is 2?
20% of 8 is what number?
30% of what number is 12?
Tia bought a sweater discounted 20% off of original price. She saved $4.60. What was orginal price of sweater?
***THINK: 20% of original price is 4.60.
The price of a video game was originally $35.60. If you purchased it for 30>26 with a coupon, what % of the
original price did you pay? ***THINK: What % of original price is what I paid.
What is % of change if orig price is 30 and final price is 90.
What is % of change if orig price is 40 and final price is 30.
There's $25 in your wallet. You have the same # of $1, $5, and quarters. How many of each type of bill or coin
do you have?
You have the same number of nickels, dimes, and pennies. You have $1.44. How many of each coin? ***UNITS
in this problem must be the same (either both in cents or both in dollars).
5 less than the difference of twice a number and 3 is 10. What's the number?
10 less than twice the difference of a number and 3 is 20. What is the number?
Henry is 3 inches more than one-half of Charlie's height. Henry is 24 inches tall. How tall is Charlie?
Bob's age is 4 times greather than Susanne's age. Dakota is three years younder than Susanne. The sum of their
ages is 93. How old is each?
Clarence's age is 3/5 of Amy's. Isiah's is half of Amy's age. The sum of their ages is 21 years. What is Isiah's
age?
Equations NOTES & Home Learning Practice: Page 5 of 8
15. The sum of the least and greatest of 3 consecutive integers is 98. What is the largest integer?
1. If two angles are complementary and one measures 3x while the other measures 2x + 5, what does each angle
measure?
17. If two angles are supplementary and one measures 3x while the other measures 2x + 5, what does each angle
measure?
18. If two angles are VERTICAL and one measures 3x while the other measures 2x + 5, what does each angle
measure?
HOME LEARNING
Solve the following equations; I recommend using cursive letters if you use variables a, b, g, j, l, q, s, t, y, z. I do NOT
recommend using e, i, o as variables.
LINEAR (1st degree) Equations
1.
2 + 5k = 3k -6
k = -4
2.
5x – 7 = x
x = 7/4
3.
3(2b-1) – 7 = 6b – 10
All real numbers are solutions
4.
6(5m-3) = 1/3 (24m + 12)
m=1
5.
5x + 5 = 3 (5x-4) –10x
No solutions
6.
3 + 2 (x-1) = 1
x=0
7.
x/5 = 7/12
x=35/12
8.
(a+7)/8 = 5
x=33
9.
-3 = 2 + x/11
x = -55
10. (3/2)a – 8 = 11
a = 38/3
11. (b-4)/6 = b/2
x = -2
12. x/2 + 1 = (1/4) x – 6
x = -28
13. 1.3c = 3.3c + 2.8
c = -1.4
Quadratic, cubic, quartic, pentic, etc. Equations
14. X2 = 100
x = 10 or x = -10
15. X3 = -125
x = -5
16. X4 = 81
x = 3 or x = -3
6
17. X = -1
No Real # solutions (no # to an even power will yield a negative product)
18. X5 = 32
x=2
Absolute Value Equations
19. |x| = 12
20. |x| = 0
21. |x| = -5
22. |x+5| = 17
23. |x+7| = 0
24. |b-1| = -3
25. |2t - 4| = 8
26. |5h + 2| = -8
27. 2|x| - 3 = 8
28. 4 - 3|x| = 10
29. 4/(|x|) + 12 = 14
x = 12 or x = -12
x=0
No solutions because || cannot be negative
x = 12 or x = -22
x = -7
No solutions because || cannot be negative
t = 6 or t = -2
No solutions because || cannot be negative
x = 11/ 2 or x = - 11/2
No solutions because || cannot be negative
x = 2 or x = -2
Literal Equations: Solve for the indicated variable
30. 4m – 3n = 8
for m
m = (8+3n)/4
31. 28 = t (r+4)
for t
t = 28 / (r+4)
32. (k-2) / 5 = 11j
for k
k = 55j + 2
33. a(q-8) = 23
for q
q = (23 + 8a) / a
34. d + 5c = 3d – 1 for d
d = (5c + 1) / 2
Equations NOTES & Home Learning Practice: Page 6 of 8
Solve the following inequalities & GRAPH the solution set; answers are in braces for you to check your work.
Remember to change the direction of your ineq symbol if you mult or div by a neg.
Linear Inequalities
35. 3a + 6 ≤ 49
36. 22 < m - 8
37. w - 5 ≥ 2w
38. -3x < 9
39. (-3/7) r > 21
40. 2y + 11 ≥ -24y
41. 4 (3t-5) + 7 ≥ 8t + 3
42. 2 (h + 6) < -3 (8 - h)
43. 7 + b ≤ 2 (b - 13) - 12
{a | a ≤ 43/3} or 14.3333... & don't forget to graph all ineqs below
{m | m > 30}
{w | w ≤ -5}
{x | x > -3}
{r | r < -49}
{y | y ≥ -11/26}
{t | t ≥ 4}
{h | h > 36}
{b | b ≥ 45}
Absolute Value Inequalities --- Remember to ISOLATE the || BEFORE solving and then use GOLA trigger; GRAPH
44. |c| < 7
{c | -7 < c < 7}
45. |d| ≥ 9
{d | d ≥ 9 or d ≤ -9}
46. |f| < 0
No solutions because || is never less than zero (AKA negative)
47. |g| ≤ 0
g=0 is only solution because || is never less than zero but can = 0
48. |j| > 0
all real numbers except 0 since || is always > 0 except when it = 0
49. |k| ≥ 0
all real numbers since || is always ≥ 0 (AKA non-negative)
50. |l| < -7
No solutions because || is never negative or less than negative
51. |n| ≥ -9
All real numbers are solutions because || is always > neg
52. |-p| ≤ 15
{p | -15 ≤ p ≤ 15}
53. -|q| ≥ 17
No solutions because || is never less than negative
54. -|r| ≥ -4
{r | -4 ≤ r ≤ 4}
55. |s + 2| < 11
{s | -13 < s < 9}
56. |t - 1| < -2
No solutions because || is never negative or less than negative
57. |3u+6| ≥12
{u | u ≥ 2 or u ≤ -6}
58. |v-6|≥-5
All real numbers are solutions because || is always > neg
59. 2|c+4| + 8 ≤ 12
{c | -6 ≤ c ≤ -2}
60. -3|z - 2| - 4 < -22
{z | z > 8 or z < -4}
Word Problems; you may use a calculator for these. Remember that “is” means “=” and that “of” means "multiply."
61. What number is 70% of 23? Answer: The number is 16.1.
62. 25% of what number is 75? Answer: The number is 300.
63. 80 is what percent of 100? Answer: It is 80%.
64. 5 times the sum of a number and 10 is 75. What is the number? Answer: The number is 5.
65. The quotient of a number and 12 is 5 less than the number. What is the number? Answer: The number is 5.45...
66. Twice the difference of a number and 3 is equivalent to the number tripled. What’s the number? Answer:
The number is -6.
67.
68.
The square of a number is 8. What’s the number? Answer: The number is the square root of 8.
What is the percent of change if the original price is $40 and the new price is $30? Answer: The % of
decrease is 25%.
99. What is the percent of change if the original price is $50 and the new price is $110? Answer: The % of
increase is 120%.
70. <A and <B are vertical angles. Vertical angles are congruent. If m<A= 2x+4 and the m<B=x+20, find the
measure of each angle. Answer: Both angles measure 36 degrees.
71. <C and <D are complementary. If m<C = x and m<D = 2x, find the measure of each angle. Answer: m<C =
30° and m<D = 60°.
Equations NOTES & Home Learning Practice: Page 7 of 8
72. <F and <G are supplementary. If m<F = 2x and m<G= 3x, what’s the measure of each angle? Answer: m<F=
72° and m<G = 108°.
73. What is the angle measure of each angle in a regular hexagon? Answer: All six angles measure 120 degrees.
74. Liz is twice as old as Sam. Sam is ½ as old as John. The sum of their ages is 100. How old is each? Answer:
John and Liz are 40 years old; Sam is 20 years old.
75. 3 consecutive odd numbers sum to 39. What are the numbers? Answer: The consec odd ints are 11, 13, 15.
Equations NOTES & Home Learning Practice: Page 8 of 8