Diagnostic Quiz Review Algebra 2 Concepts

In this Presentation, we will review: Solve Multi-Step Equations Distributive Property Solving inequalities Graphing Inequalities Compound Inequalities

Linear Equations with 2 Functions Sometimes linear equations will have more than one operation In this instance operations are defined as add, subtract, multiply or divide The rule is First you add or subtract Second you multiply or divide. Then, deal with exponents, followed by parenthesis It’s Order of Operations BACKWARDS!!!

Example # 1 2x + 6 = 16 In this case the problem says 2 times a number plus 6 is equal to 16 The first step is to add or subtract. To undo the addition of 6 I have to subtract 6 from both sides which looks like this: 16 minus 6 equals 10 The 6’s on the left side cancel each other out 2x + 6 = 16 - 6 -6 2x = 10 What is left is a one step equation 2x = 10

Example #1 (Continued) 2x = 10 The second step is to multiply or divide To undo the multiplication here you would divide both sides by 2 which looks like this: 10 divided by 2 is equal to 5 2x = 10 The 2’s on the left cancel out leaving x 2 2 x = 5 The solution to the equation 2x + 6 = 16 is x = 5

Combining Like Terms First When solving multi-step equations, sometimes you have to combine like terms first. The rule for combining like terms is that the terms must have the same variable and the same exponent. Example: You can combine x + 5x to get 6x You cannot combine x + 2x2 because the terms do not have the same exponent

Example # 2 7x – 3x – 8 = 24 7x – 3x – 8 = 24 4x – 8 = 24 I begin working on the left side of the equation On the left side I notice that I have two like terms (7x, -3x) since the terms are alike I can combine them to get 4x. 7x – 3x – 8 = 24 4x – 8 = 24 After I combine the terms I have a 2-step equation. To solve this equation add/subtract 1st and then multiply/divide

Example # 2 (continued) 4x – 8 = 24 4x – 8 = 24 +8 +8 4x = 32 Step 1: Add/Subtract Since this equation has – 8, I will add 8 to both sides 4x – 8 = 24 +8 +8 24 + 8 = 32 The 8’s on the left cancel out 4x = 32 I am left with a 1-step equation

The solution that makes the statement true is x = 8 Example # 2 (continued) 4x = 32 Step 2: Multiply/Divide In this instance 4x means 4 times x. To undo the multiplication divide both sides by 4 32  4 = 8 The 4’s on the left cancel out leaving x 4x = 32 4 4 x = 8 The solution that makes the statement true is x = 8

Solving equations using the Distributive Property When solving equations, sometimes you will need to use the distributive property first. At this level you are required to be able to recognize and know how to use the distributive property Essentially, you multiply what’s on the outside of the parenthesis with EACH term on the inside of the parenthesis Let’s see what that looks like…

Example #3 5x + 3(x +4) = 28 5x + 3(x +4) = 28 In this instance I begin on the left side of the equation I recognize the distributive property as 3(x +4). I must simplify that before I can do anything else 5x + 3(x +4) = 28 5x +3x +12 = 28 After I do the distributive property I see that I have like terms (5x and 3x) I have to combine them to get 8x before I can solve this equation

I am now left with a 2-step equation Example # 3(continued) 8x + 12 = 28 I am now left with a 2-step equation Step 1: Add/Subtract The left side has +12. To undo the +12, I subtract 12 from both sides 8x + 12 = 28 28 – 12 = 16 The 12’s on the left cancel out leaving 8x -12 -12 8x = 16

Example # 3 (continued) 8x = 16 8x = 16 8 8 x = 2 Step 2: Multiply/Divide On the left side 8x means 8 times x. To undo the multiplication I divide both sides by 8 8x = 16 16  8 = 2 The 8’s on the left cancel out leaving x 8 8 x = 2 The solution that makes the statement true is x = 2

Distributing a Negative Distributing a negative number is similar to using the distributive property. However, students get this wrong because they forget to use the rules of integers Quickly the rules are…when multiplying, if the signs are the same the answer is positive. If the signs are different the answer is negative

Example #4 4x – 3(x – 2) = 21 I begin by working on the left side of the equation. In this problem I have to use the distributive property. However, the 3 in front of the parenthesis is a negative 3. When multiplying here, multiply the -3 by both terms within the parenthesis. Use the rules of integers 4x – 3(x – 2) = 21 4x – 3x + 6 = 21 After doing the distributive property, I see that I can combine the 4x and the -3x to get 1x or x

The 6’s cancel out leaving x Example # 4 (continued) 4x – 3x + 6 = 21 x + 6 = 21 After combining like terms you are left with a simple one step equation. To undo the +6 subtract 6 from both sides of the equation x + 6 = 21 21 – 6 = 15 The 6’s cancel out leaving x -6 -6 x = 15 The solution is x = 15

Multiplying by a Reciprocal First Sometimes when doing the distributive property involving fractions you can multiply by the reciprocal first. Recall that the reciprocal is the inverse of the fraction and when multiplied their product is equal to 1. The thing about using the reciprocal is that you have to multiply both sides of the equation by the reciprocal. Let’s see what that looks like…

Example # 5 In this example you could distribute the 3/10 to the x and the 2. The quicker way to handle this is to use the reciprocal of 3/10 which is 10/3 and multiply both sides of the equation by 10/3 On the left side of the equation, after multiplying by the reciprocal 10/3 you are left with 120/3 which can be simplified to 40 On the right side of the equation the reciprocals cancel each other out leaving x + 2 The new equation is: 40 = x + 2

Example #5 (continued) 40 = x + 2 After using the reciprocals you are left with a simple one-step equation To solve this equation begin by working on the right side and subtract 2 from both sides of the equation The 2’s on the right side cancel out leaving x 40 = x + 2 40 – 2 = 38 - 2 -2 38 = x The solution to the equation is x = 38

Your Turn 6x – 4(9 –x) = 106 14.2 2x + 7 = 15 4 6 = 14 – 2x 8 1.5

Your Turn 3 -52 -13 -1 5 5m – (4m – 1) = -12 55x – 3(9x + 12) = -64 6. 7. 8. 5m – (4m – 1) = -12 9. 55x – 3(9x + 12) = -64 10. 9x – 5(3x – 12) = 30

Greater than or equal to Inequality Symbols Less than Not equal to Less than or equal to Greater than or equal to Greater than

Transformations for Inequalities Add/subtract the same number on each side of an inequality Multiply/divide by the same positive number on each side of an inequality If you multiply or divide by a negative number, you MUST flip the inequality sign!

Ex: Solve the inequality. +3 +3 2x<11 x< Flip the sign after dividing by the -3!

Graphing Linear Inequalities Remember: < and > signs will have an open dot o and signs will have a closed dot graph of graph of 4 5 6 7 -3 -2 -1

Example: Solve and graph the solution. 6 7 8 9

Compound Inequality An inequality joined by “and” or “or”. Examples think between think oars on a boat -3 -2 -1 0 1 2 3 4 5 -4 -3 -2 -1 0 1 2

What is the difference between and and or? AND means intersection -what do the two items have in common? OR means union -if it is in one item, it is in the solution A B

● ● ● 1) Graph x < 4 and x ≥ 2 a) Graph x < 4 o o b) Graph x ≥ 2 3 4 2 o 3 4 2 o b) Graph x ≥ 2 3 4 2 ● ● c) Combine the graphs d) Where do they intersect? ● 3 4 2 o

● ● 2) Graph x < 2 or x ≥ 4 a) Graph x < 2 o o b) Graph x ≥ 4 3 4 2 o 3 4 2 o b) Graph x ≥ 4 3 4 2 ● 3 4 2 ● c) Combine the graphs

3) Which inequalities describe the following graph? -2 -1 -3 o y > -3 or y < -1 y > -3 and y < -1 y ≤ -3 or y ≥ -1 y ≥ -3 and y ≤ -1

4) Graph the compound inequality 6 < m < 8 When written this way, it is the same thing as 6 < m AND m < 8 It can be rewritten as m > 6 and m < 8 and graphed as previously shown, however, it is easier to graph everything between 6 and 8! 7 8 6 o

5) Which is equivalent to -3 < y < 5? y > -3 or y < 5 y > -3 and y < 5 y < -3 or y > 5 y < -3 and y > 5

6) Which is equivalent to x > -5 and x ≤ 1?

● ● 7) 2x < -6 and 3x ≥ 12 o o o o -3 -6 -3 -6 4 7 1 4 7 1 Solve each inequality for x Graph each inequality Combine the graphs Where do they intersect? They do not! x cannot be greater than or equal to 4 and less than -3 No Solution!! -3 -6 o -3 -6 o 4 7 1 o ● 4 7 1 o ●

8) Graph 3 < 2m – 1 < 9 Remember, when written like this, it is an AND problem! 3 < 2m – 1 AND 2m – 1 < 9 Solve each inequality. Graph the intersection of 2 < m and m < 5. 5 -

9) Graph x < 2 or x ≥ 4 5 -

The whole line is shaded!! 10) Graph x ≥ -1 or x ≤ 3 The whole line is shaded!!

-9 < t+4 < 10 -13 < t < 6 Think between! Example: Solve & graph. -9 < t+4 < 10 -13 < t < 6 Think between! -13 6

Last example! Solve & graph. -6x+9 < 3 or -3x-8 > 13 -6x < -6 -3x > 21 x > 1 or x < -7 Flip signs Think oars -7 1