Goals and Objectives Life throws at us many different types of - PDF document

Slide 1 / 224 Slide 2 / 224 New Jersey Center for Teaching and Learning Algebra II Progressive Mathematics Initiative This material is made freely available at www.njctl.org and is intended for the non-commercial use of Fundamental Skills of students and teachers. These materials may not be used for any commercial purpose without the written permission of the owners. NJCTL maintains its Algebra website for the convenience of teachers who wish to make their work available to other teachers, participate in a virtual professional learning 2013-09-11 community, and/or provide access to course materials to parents, students and others. www.njctl.org Click to go to website: www.njctl.org Slide 3 / 224 Slide 4 / 224 click on the topic to go to that section Table of Contents Solving Equations and Inequalities Solving Equations and Factoring Inequalities Exponents Radicals Return to Table of Contents Slide 5 / 224 Slide 6 / 224 Solving Equations and Inequalities Solving Equations and Inequalities Why do we need this? Goals and Objectives Life throws at us many different types of problems. Being able to solve these Students will be able to solve a wide variety of equations and inequalities. problems makes us successful. Developing logic skills that we can apply in different situations helps us solve those problems with ease.

Slide 7 / 224 Slide 8 / 224 Solving Equations and Inequalities Solving Equations and Inequalities Remember... Steps for solving equations: There are typically two "sides" to an equation. They · 1) Remove any parentheses. are separated by the equals sign. 2) Collect like terms. An equation is like a balance. What you do to one · 3) Move all variables to one side. side, must be done to the other side! 4) Move all constants to one side. Use opposite operations (+/-, x/÷) when moving · 5) Remove any coefficient. 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 9 / 224 Slide 10 / 224 Solving Equations and Inequalities Solving Equations and Inequalities 1 Try on your own. Use the rules from your notes to help you. The nice part about how equations work is that you can always check your answer. Plug in your result to the original equation -3(2x + 4) - 8 = 2(x - 5) - 2 even if your first answer was incorrect. What happens? Plug in incorrect answer: Plug in correct answer: -3(2x + 4) - 8 = 2(x - 5) - 2 -3(2x + 4) - 8 = 2(x - 5) - 2 Slide 11 / 224 Slide 12 / 224 Solving Equations and Inequalities Solving Equations and Inequalities More examples: More examples: -2(a + 3) - 2 = 3a - 15 - 8 3m - 4 + 2m = 5 + 4m

Slide 13 / 224 Slide 14 / 224 Solving Equations and Inequalities Solving Equations and Inequalities and a few more... Will all of our answers be nice, whole numbers? and a few more... Will all of our answers be nice, whole numbers? 3(x - 4) + 5 = 1 + 7(x - 2) -4y - 5 + 3y - 6 = 4 + 2y + 7 Slide 15 / 224 Slide 16 / 224 Solving Equations and Inequalities Solving Equations and Inequalities 2 Solve for x: 2(x - 4) - 3(x + 5) = 2x - 8 3 Find m: 5m - 4 - 2m = 4m - 12 + 6 Slide 17 / 224 Slide 18 / 224 Solving Equations and Inequalities Solving Equations and Inequalities 4 Solve the equation: -6(y + 2) = -7(y - 3) 5 Find x: 5 - 2x - 4 - 4x = -3 - 5x + 2x - 5

Slide 19 / 224 Slide 20 / 224 Solving Equations and Inequalities Solving Equations and Inequalities 6 Solve for x: -4 - (x - 5) = 2 - (3x - 8) Try: -2 - (m - 3) = -(m - 4) - 2 Slide 21 / 224 Slide 22 / 224 Solving Equations and Inequalities Solving Equations and Inequalities For example: Sometimes, the variable cancels out and you are left with a numerical statement. If that happens, you will have one of two -p + 3 = -p + 6 b - 8 = b - 8 possible answers. results in 3 ≠ 6 results in -8 = -8 True Statement = infinitely many solutions This is a false This is a true False Statement = no solution statement and statement and there is no there are solution. infinitely many solutions. Slide 23 / 224 Slide 24 / 224 Solving Equations and Inequalities Solving Equations and Inequalities 7 What is the solution to the equation: 5 - 2(x + 3) = -2x + 4 8 Find x: -2(x - 3) + 4 = -x + 10 A -4 A 10 B 0 B 0 C No solution C No solution D Infinitely many solutions D Infinitely many solutions

Slide 25 / 224 Slide 26 / 224 Solving Equations and Inequalities Solving Equations and Inequalities 9 Solve: m - 6 - 3 - 2m = m - 4 - m - 5 - m 10 Find y: -4y + 8 - y = y - 6 - 8 A 9 A 11/3 B 0 B 0 C No solution C No solution D Infinitely many solutions D Infinitely many solutions Slide 27 / 224 Slide 28 / 224 Solving Equations and Inequalities Solving Equations and Inequalities 11 Find the answer to the equation: -2(x - 4) + 5(x + 3) = 3x - 12 + 2 Now for fractions... A 4 B 0 C No solution D Infinitely many solutions Slide 29 / 224 Slide 30 / 224 Solving Equations and Inequalities Solving Equations and Inequalities 12 Solve: Try:

Slide 31 / 224 Slide 32 / 224 Solving Equations and Inequalities Solving Equations and Inequalities 14 Solve the equation: 13 Find x: Slide 33 / 224 Slide 34 / 224 Solving Equations and Inequalities Solving Equations and Inequalities 15 Find m: 16 Solve the equation: Slide 35 / 224 Slide 36 / 224 Solving Equations and Inequalities Solving Equations and Inequalities Solving formulas for specific variables Try...this one is a bit tougher... incorporates all of these rules. Try... 4pc = 2t - 9dm solve for d

Slide 37 / 224 Slide 38 / 224 Solving Equations and Inequalities 17 Solve the following formula for C. A D B E C Slide 39 / 224 Slide 40 / 224 Solving Equations and Inequalities Solving Equations and Inequalities 18 Solve for h: 19 Solve for h: A A D D B B E E C C Slide 41 / 224 Slide 42 / 224 Solving Equations and Inequalities Solving Equations and Inequalities 20 Solve for w: 21 Solve for m: A A D D B B E E C C

Slide 43 / 224 Slide 44 / 224 Solving Equations and Inequalities Solving Equations and Inequalities Remember inequalities? 22 Solve for x: [ , ] ( , ) >, < ≥, ≤ A D B All symbols indicate All symbols indicate E solutions will not solutions will C include referenced include referenced points. points. Slide 45 / 224 Slide 46 / 224 Solving Equations and Inequalities Solving Equations and Inequalities You solve inequalities the same way that you solve equations. The Solutions to equations are single points. Solutions to inequalities are only difference is that you flip an inequality symbol if you multiply or regions of points. divide by a negative number. Solve: 3m + 4 - 5m < 3m + 6 Draw a graph to represent the solution to the last problem: m > -2/5 Slide 47 / 224 Slide 48 / 224 Solving Equations and Inequalities Solving Equations and Inequalities Try solving and graphing the inequality... One more, solve and graph the inequality... 2x - 6 + 3 ≥ 5x - 4 3(x - 2) - 4(x + 3) ≤ -2(x - 4)

Slide 49 / 224 Slide 50 / 224 Solving Equations and Inequalities Solving Equations and Inequalities 23 Solve for x: 5(x - 3) + 2 -(2x - 4) 24 Solve the following inequality: 6y - 2 - 2y > 3y + 5 A y > 7/9 x ≥ 17/2 A B y < 7/9 x ≤ 17/7 B C y > 7/5 D y > 3/5 C x ≥ 21/5 E y > 7 D x ≤ 21/5 E x ≤ 14/5 Slide 51 / 224 Slide 52 / 224 Solving Equations and Inequalities Solving Equations and Inequalities 25 Find values for p: 2(p + 3) - 5 4(p + 4) 26 Which of the following numbers would be a solution to the following inequality? 3p + 6 - 5p > 10p - 4 - 8 p ≥ -15/2 A A 0 B 5/2 B p ≤ -15/2 C 5 C p ≥ -5/2 D 15/4 E 9 D p ≤ -5/2 E Answer not listed. Slide 53 / 224 Slide 54 / 224 Solving Equations and Inequalities 27 Solve for n: 4 - 3(n - 3) 2(n + 4) - 2 A n ≤ -11/5 B n ≥ -11/5 n ≥ 19/5 C Factoring D n ≤ 19/5 E Answer not listed. Return to Table of Contents

Slide 55 / 224 Slide 56 / 224 Factoring Factoring Why do we need this? Goals and Objectives The more we can simplify a problem, the Students will be able to factor complex easier it is to solve. Factoring allows us to expressions and solve equations break up expressions into smaller parts using factoring. and, much of the time, simplify our strategies to solve them. Slide 57 / 224 Slide 58 / 224 Factoring Factoring Multiply the following: Factoring is undoing what you just did. It breaks up expressions into parts and pieces. 3x(2x 2 + 5) (x + 3)(x - 9) (2x - 1)(3x - 4) Slide 59 / 224 Slide 60 / 224 Factoring Factoring Factor out the GCF (greatest common factor). You can easily check your answer by distributing the GCF back to all of your terms. 5a 2 b - 10ab 12a 3 b 2 + 4ab 2 8x + 32y Factor then check your answer: 3x 2 y 2 + 12xy 2 + 6y 2 6m 4 n 3 + 18m 3 n 2 - 36m 2 n