This material is made freely available at www.njctl.org and is intended for the non-commercial use of students and teachers. These materials may not be used for any commercial purpose without the written permission of the owners. NJCTL maintains its website for the convenience of teachers who wish to make their work available to other teachers, participate in a virtual professional learning community, and/or provide access to course materials to parents, students and others. Click to go to website: www.njctl.org New Jersey Center for Teaching and Learning Progressive Mathematics Initiative Slide 1 / 224 Algebra II Fundamental Skills of Algebra www.njctl.org 2013-09-11 Slide 2 / 224 Table of Contents Solving Equations and Inequalities Factoring Exponents Radicals click on the topic to go to that section Slide 3 / 224
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
This material is made freely available at www.njctl.org and is intended for the non-commercial use of students and teachers. These materials may not be used for any commercial purpose without the written permission of the owners. NJCTL maintains its website for the convenience of teachers who wish to make their work available to other teachers, participate in a virtual professional learning community, and/or provide access to course materials to parents, students and others.
1) Remove any parentheses.2) Collect like terms.3) Move all variables to one side.4) Move all constants to one side.5) Remove any coefficient.
Slide 7 / 224
Solving Equations and Inequalities
Remember...
· There are typically two "sides" to an equation. They are separated by the equals sign.
· An equation is like a balance. What you do to one side, must be done to the other side!
· Use opposite operations (+/-, x/÷) when moving variables and constants across the equals sign.
· "Collecting like terms" is just organizing and simplifying each side of the equation. Do not use opposites when collecting like terms.
Slide 8 / 224
1 Try on your own. Use the rules from your notes to help you.
Solving Equations and Inequalities
-3(2x + 4) - 8 = 2(x - 5) - 2
Slide 9 / 224
Solving Equations and Inequalities
The nice part about how equations work is that you can always check your answer. Plug in your result to the original equation even if your first answer was incorrect. What happens?
6 Solve for x: -4 - (x - 5) = 2 - (3x - 8)Solving Equations and Inequalities
Slide 19 / 224
Try:
Solving Equations and Inequalities
-2 - (m - 3) = -(m - 4) - 2
Slide 20 / 224
Solving Equations and Inequalities
Sometimes, the variable cancels out and you are left with a numerical statement. If that happens, you will have one of two possible answers.
True Statement = infinitely many solutionsFalse Statement = no solution
Slide 21 / 224
For example:
-p + 3 = -p + 6
results in 3 ≠ 6
This is a false statement and
there is no solution.
b - 8 = b - 8
results in -8 = -8
This is a true statement and
there are infinitely many
solutions.
Solving Equations and Inequalities
Slide 22 / 224
7 What is the solution to the equation: 5 - 2(x + 3) = -2x + 4
A -4B 0C No solutionD Infinitely many solutions
Solving Equations and Inequalities
Slide 23 / 224
8 Find x: -2(x - 3) + 4 = -x + 10
A 10B 0C No solutionD Infinitely many solutions
Solving Equations and Inequalities
Slide 24 / 224
9 Solve: m - 6 - 3 - 2m = m - 4 - m - 5 - m
A 9B 0C No solutionD Infinitely many solutions
Solving Equations and Inequalities
Slide 25 / 224
10 Find y: -4y + 8 - y = y - 6 - 8
A 11/3B 0C No solutionD Infinitely many solutions
Solving Equations and Inequalities
Slide 26 / 224
11 Find the answer to the equation: -2(x - 4) + 5(x + 3) = 3x - 12 + 2
A 4B 0C No solutionD Infinitely many solutions
Solving Equations and Inequalities
Slide 27 / 224
Solving Equations and Inequalities
Now for fractions...
Slide 28 / 224
Try:
Solving Equations and Inequalities
Slide 29 / 224
12 Solve:
Solving Equations and Inequalities
Slide 30 / 224
13 Find x:
Solving Equations and Inequalities
Slide 31 / 224
14 Solve the equation:
Solving Equations and Inequalities
Slide 32 / 224
15 Find m:
Solving Equations and Inequalities
Slide 33 / 224
16 Solve the equation:
Solving Equations and Inequalities
Slide 34 / 224
Solving Equations and Inequalities
Solving formulas for specific variables incorporates all of these rules. Try...
4pc = 2t - 9dm solve for d
Slide 35 / 224
Solving Equations and Inequalities
Try...this one is a bit tougher...
Slide 36 / 224
Slide 37 / 224
17 Solve the following formula for C.
A
B
C
D
E
Solving Equations and Inequalities
Slide 38 / 224
18 Solve for h:
A
B
C
D
E
Solving Equations and Inequalities
Slide 39 / 224
19 Solve for h:
A
B
C
D
E
Solving Equations and Inequalities
Slide 40 / 224
20 Solve for w:
A
B
C
D
E
Solving Equations and Inequalities
Slide 41 / 224
21 Solve for m:
A
B
C
D
E
Solving Equations and Inequalities
Slide 42 / 224
22 Solve for x:
A
B
C
D
E
Solving Equations and Inequalities
Slide 43 / 224
Solving Equations and Inequalities
Remember inequalities?
≥, ≤ >, <[ , ] ( , )
All symbols indicate solutions will include referenced points.
All symbols indicate solutions will not include referenced points.
Slide 44 / 224
Solving Equations and Inequalities
You solve inequalities the same way that you solve equations. The only difference is that you flip an inequality symbol if you multiply or divide by a negative number.
Solve: 3m + 4 - 5m < 3m + 6
Slide 45 / 224
Solutions to equations are single points. Solutions to inequalities are regions of points.
Solving Equations and Inequalities
Draw a graph to represent the solution to the last problem: m > -2/5
Slide 46 / 224
Solving Equations and Inequalities
Try solving and graphing the inequality...
2x - 6 + 3 ≥ 5x - 4
Slide 47 / 224
One more, solve and graph the inequality...
Solving Equations and Inequalities
3(x - 2) - 4(x + 3) ≤ -2(x - 4)
Slide 48 / 224
23 Solve for x: 5(x - 3) + 2 -(2x - 4)
A
B
CDE
Solving Equations and Inequalities
x ≤ 17/7
x ≥ 17/2
x ≤ 21/5x ≤ 14/5
x ≥ 21/5
Slide 49 / 224
24 Solve the following inequality: 6y - 2 - 2y > 3y + 5
A y > 7/9B y < 7/9C y > 7/5D y > 3/5E y > 7
Solving Equations and Inequalities
Slide 50 / 224
25 Find values for p: 2(p + 3) - 5 4(p + 4)
A
B
C
D
E Answer not listed.
Solving Equations and Inequalities
p ≥ -15/2
p ≤ -15/2
p ≥ -5/2
p ≤ -5/2
Slide 51 / 224
26 Which of the following numbers would be a solution to the following inequality? 3p + 6 - 5p > 10p - 4 - 8
To factor the difference of squares, the difference of cubes and the sum of cubes, use the following formulas:
Factoring
a2 - b2 ⇒ (a - b)(a + b)
a3 - b3 ⇒ (a - b)(a2 + ab + b2)
a3 + b3 ⇒ (a + b)(a2 - ab + b2)
Slide 91 / 224
Try...
Factoring
4p2 - q2 16m2 - 1 64p3 + y3
a2 - b2 ⇒ (a - b)(a + b)a3 - b3 ⇒ (a - b)(a2 + ab + b2)a3 + b3 ⇒ (a + b)(a2 - ab + b2)
Slide 92 / 224
Factoring
a2 - b2 ⇒ (a - b)(a + b)a3 - b3 ⇒ (a - b)(a2 + ab + b2)a3 + b3 ⇒ (a + b)(a2 - ab + b2)Factor...
25x2 - 81y2 x3y3 + 1 8m3 - 125n3
Slide 93 / 224
46 Factor:
A (11m - 10n)(11m + 10m) B (121m - n)(m + 100n) C (11m - n)(11m + 100n) D Not factorable E Solution not shown
121m2 + 100n2
Factoring
Slide 94 / 224
47 Factor:
A (3m - 8np)(3m + 8np) B (3m + 4np)(3m2 - 12mnp + 4n2p2) C (3m + 4np)(9m2 - 12mnp + 16n2p2) D Not factorable E Solution not shown
Factoring
27m3 + 64n3p3
Slide 95 / 224
48 Factor:
A (a - 5b)(a + 5b) B (a - 5b)(a2 + 5ab + 5b2) C (a + 5b)(a2 - 5ab + 25b2) D Not FactorableE Solution not shown
Factoring
a3 - 125b3
Slide 96 / 224
49 Factor:
A (m + n)(m - n) B (m + n)(m + n) C (m - n)(m - n) D Not factorableE Solution not shown
Factoring
m2 - n2
Slide 97 / 224
50 Factor:
A (d - 1)(d + 1) B (d - 1)(d2 + d + 1) C (d + 1)(d2 - d + 1) D Not factorableE Solution not shown
Factoring
d3 - 1
Slide 98 / 224
51 Factor:
A (3m + 2n)(3m - 2n) B (3m + 2n)(9m2 - 6mn + 4n2) C (3m - 2n)(9m2 + 6mn + 4n2) D Not factorableE Solution not shown
Factoring
27m3 + 4n3
Slide 99 / 224
52 Factor:
A (16x - 2y)(16x + 2y) B (6x - 2y)(6x2 + 12xy + 2y2) C (6x - 2y)(36x2 + 12xy + 4y2) D Not factorableE Solution not shown
Factoring
216x3 - 8y3
Slide 100 / 224
Factoring
Factoring by Grouping.
What happens when there are 4 terms?
4ap - 4a + 3xp - 3x
Slide 101 / 224
Factoring
Try... xy + 4x - 3y - 12
Slide 102 / 224
Factoring
Two more...pq + 4p + 3q + 12 mn - pm - qn + qp
Slide 103 / 224
53 Factor by grouping:
A (r + 4)(s - 7) B (r - 7)(s + 4) C (rs - 7)(rs + 4) D Not factorableE Solution not shown
Factoring
rs - 7r + 4s - 28
Slide 104 / 224
54 Factor:
A (n - 3)(m + 4n) B (n - 3)(m - 4n) C (n + 4)(m - n) D Not factorableE Solution not shown
Factoring
mn + 3m - 4n2 - 12n
Slide 105 / 224
55 Factor:
A (3k - 2)(g + 6) B (3g + 2)(k - 6) C (3k - 2)(g + 6) D Not factorableE Solution not shown
Factoring
3gk - 18g + 2k - 12
Slide 106 / 224
56 Factor by grouping:
A (m - 4)(2p - 7) B (m + 7)(2p + 4) C (m - 4)(2p + 7) D Not factorableE Solution not shown
Factoring
2mp - 8p - 7m + 28
Slide 107 / 224
57 Factor by grouping
A (x + 4)(2y + 3) B (x + 4)(2y - 3) C (x - 3)(2y + 4) D Not factorableE Solution not shown
2xy + 3x + 8y - 12
Factoring
Slide 108 / 224
58 Factor:
A (3m - 5)(p + 2n) B (3m + 5)(p - 2n) C (3m - 2n)(p + 5) D Not factorableE Solution not shown
Factoring
3mp + 15m - 2np - 10n
Slide 109 / 224
Now, let's combine all of the situations. In any factoring problem, factor out the GCF first.
Factoring
Factor these completely...
2x3 - 22x2 + 48x 3m3n + 3m2n - 18mn
Slide 110 / 224
Factoring
Factor completely...
4x3 - 32y3 54a4 + 2ab3
Slide 111 / 224
59 Factor completely:
A -3mn(2m - 1)(m - 3) B 3mn(2m + 1)(m - 3) C -3n(2m - 1)(m2 + 3m + 9) D Not factorableE Solution not shown
Factoring
-6m3n + 21m2n - 9mn
Slide 112 / 224
60 Factor completely:
A -3(2p + 5)(4p3 - 10p + 25) B -3p(16p2 + 25) C -3p(4p - 5)(4p - 5) D Not factorableE Solution not shown
Factoring
-48p3 + 75p
Slide 113 / 224
61 Factor completely:
A (4p2 - 3)(m - 4) B 4p2(pm - 4)(pm - 3) C 4p2m(m - 4)(p + 3) D Not factorableE Solution not shown
Factoring
4p3m - 12p3 - 16mp2 + 48p2
Slide 114 / 224
62 Factor completely:
A 2xy(9x - 1) B 2xy(3x - 1)(3x + 1) C 2y(9x2 - x) D Not factorableE Solution not shown
Factoring
18x3y - 2xy
Slide 115 / 224
63 Factor completely:
A -8ab(a2 + 4) B -4ab(2a2 + a + 3)C -4ab(2a + 3)(a - 1)D Not factorableE Solution not shown
Factoring
-8a3b - 4a2b + 12ab
Slide 116 / 224
64 Factor completely:
A 2b(4b - 1)(3b + 2) B 4b(6b2 + 6b - 1) C 2b(4b + 1)(3b - 2) D Not factorableE Solution not shown
24b3 + 10b2 - 4b
Factoring
Slide 117 / 224
Factoring is often use to solve equations that are in polynomial form.
Factoring
Steps: 1) Move all terms to one side of the equation. (the other side becomes zero) 2) Factor the resulting polynomial. 3) Set each factor equal to zero. 4) Solve each equation. 5) Write the answers clearly.
Slide 118 / 224
Factoring
Solve the equation by factoring:
x2 = 9x - 18
Slide 119 / 224
One more...
Factoring
6x3 + 10x2 = 4x
Slide 120 / 224
Another example...
Factoring
6m2 = 9m - 24m3
Slide 121 / 224
65 Which of the following are solutions to the equation?
A 0B -1C -3/2D 3/2E -4F 4G -4/3H 4/3
Factoring
16x3 - 36x = 0
Slide 122 / 224
66 Find all of the solutions to:
A 0B -1C 1D -3E 3F -4G 4H 12
Factoring
-3m3 + 3m2 = -36m
Slide 123 / 224
67 Solve for p:
A 0B -1/2C 1/2D -2/5E 2/5F -4/3G -5/2H 5/2
Factoring
29p2 + 10p = -10p3
Slide 124 / 224
68 Find the values for n:
A 0B -1/2C 1/2D -1/3E -2/3F 2/3G -4H 4
Factoring
18n4 + 48n2 = 84n3
Slide 125 / 224
69 Find x:
A 0B -1/4C 1/4D -1/3E 1/3F -1/2G 1/2H 3
Factoring
6x4 = 5x3 - x2
Slide 126 / 224
70 Solve by factoring:
A 0B -1/4C 1/4D -1/2E 1/2F 1G -2H 2
Factoring
p2 + p = 2p3
Slide 127 / 224
Exponents
Return to Table of Contents
Slide 128 / 224
Exponents
Goals and ObjectivesStudents will be able to simplify complex expressions containing exponents.
Students will be able to put problems in simplest radical form, as well as be able to add, subtract, multiply and divide radical expressions.
Slide 178 / 224
Radicals
Why do we need this?Engineers and Scientists perform complicated
calculations. If a problem requires multiple operations and answers are rounded and reused for the next step,
what happens to the answer in the end? It is not as accurate as possible. In science, we round only at the end. This way, we can have more accurate answers.
Being able to work with radicals, keeping them in simplest radical form, will allow us to maintain exact
numbers as we work with problems.
Slide 179 / 224
Slide 180 / 224
Radicals
= 4.898979485566356....
Since 24 is not a perfect square, you must round the decimal to make the number reasonable. Your answer is then not an exact number. Simplest radical form allows us to simplify
Use these numbers to factor the radical. Then simplify.
Try...
Slide 182 / 224
Slide 183 / 224
Slide 184 / 224
Slide 185 / 224
98 Find:
Radicals
Slide 186 / 224
Slide 187 / 224
100 Put in simplist radical form:
A
BCD Solution not shown
Radicals
Slide 188 / 224
101 Simplify:
A
B
C
D Solution not shown
Radicals
Slide 189 / 224
102 Simplify:
A
B
C
D Solution not shown
Radicals
Slide 190 / 224
Slide 191 / 224
104 Simplify:
A
B
C
D Solution not shown
Radicals
Slide 192 / 224
Simplify: 3x + 4x and 3x + 4y
Think about: and
Adding and subtracting radicals - relate it to what you know...Radicals
Slide 193 / 224
Adding and subtracting radicals is the same as adding and subtracting terms with variables. If the roots do not match, you
cannot put them together.
Radicals
Try...
Slide 194 / 224
Slide 195 / 224
Radicals
Can you simplify these?
Slide 196 / 224
Radicals
Try...
Slide 197 / 224
Slide 198 / 224
Slide 199 / 224
106 Simplify:
A
B
C
D Solution not shown
Radicals
Slide 200 / 224
107 Put in simplest radical form and collect like terms:
A
B
C
D Solution not shown
Radicals
Slide 201 / 224
Slide 202 / 224
Slide 203 / 224
Radicals
Multiplying Radicals
Rule: Radical times radical, whole number times whole number.
Work out:
**All answers must be left in simplest radical form.
. .
Slide 204 / 224
Radicals
*Remember to leave your answers in simplest radical form
. .
Try...
Slide 205 / 224
And...
Radicals
. .
Slide 206 / 224
Slide 207 / 224
Slide 208 / 224
112 Work out:
A
B
C
D Solution not shown
Radicals
.
Slide 209 / 224
113 Multiply:
A
B
C
D Solution not shown
Radicals
.
Slide 210 / 224
114 Multiply:
A
B
C
D Solution not shown
Radicals
Slide 211 / 224
Slide 212 / 224
116 Simplify:
A
B
C
D Solution not shown
Radicals
.
Slide 213 / 224
Radicals
Dividing Radicals
Rules are the same for dividing radicals as with multiplying radicals: Radical divided by radical, whole number divided by whole number.
Slide 214 / 224
Radicals
Dividing radicals is a bit tougher because each problem is different. You know you are done with a question when there is no radical in the denominator and any fraction is reduced.
Removing a radical from the denominator is called "rationalizing the denominator."
Slide 215 / 224
Radicals
Again, each division problem is different. Try simplifying everything you can before rationalizing the denominator.