Solving Complex Equations and Inequalities in Algebra 1

1. Complex Equationslike 3(x-7) + 2(x+3)=90And InequalitiesPart 1 Alegbra 1 Standard 4

2. The California Standard • The California State board of education, in order to be in sync with national, and international standards, has set this text as their Algebra 1 standard, number 4. • “Students simplify expressions before solving equations and inequalities in one variable, such as 3(2x-5) + 4(x-2) = 12.”

3. Let’s do it. • Here we will present what this standard means and how to do problems of this nature. Here are some more examples you should be able to solve by the end of your work in this standard: • 3(x-3)-8(3x+3)=290 • 3(x-3)-8(3x+3)=290 • 7(n+3)+4(n+2)=100 • 7(n+3)+4(n+2)=100 • 3(n+4)+8(n+6)<60 • 10(n+1)+9(n+1)>900

4. Don’t be scared • Yes, we are kicking it up a notch here. • But these problems can all be solved. • Just go step by step. • Use all of the techniques that you already know plus a few more.

5. First, Remember the GOAL. • Our goal is to get the variable, to be positive, and by itself on one side of the equals sign, or the inequality sign. • And to move everything else, numbers and math operations, to the OTHER side. • We will have a solution when we have something clean and simple like: • X = 3

6. It will be a long road • Getting the variable clean and alone with take many steps with these problems. • DO NOT try to think about what the solution will be by just looking at the problem. Just look for that FIRST Step. • That one number, or one opportunity to shift things around so that it gets just a little bit closer to solution with each step.

7. Pep talk over let’s dig in: • Consider the problem: • 5(3 - 4x) + 14x = 7(2 - 5x) • A big mess as it stands, I admit. Let’s clean it up. • Remember your “order of operations”? • Our order of operations would have us to do what is in the parenthesis first, then multiply, then subtract.

8. 5(3 - 4x) + 14x = 7(2 - 5x) • But the variable x here makes it impossible to do what is in the parenthesis first. • What is needed is a way to get rid of the parenthesis so that we can move to a solution. • There is law of mathematics called “the distributive property” and it can be very helpful here.

9. The Distributive Property • a(b + c) = ab + ac • This says, in English: “If you multiply any number (call it a) by any two numbers being added, in parenthesis, call them b and c, you will get the same result as if you first multiplied a times b, and then added it to b times c.”

10. It works, a couple of examples: a ( b + c ) = ab + ac  The format 3 ( 4 + 5) = 3(4) + 3(5)  a matching problem 3 * (9) = 12 + 15 27 = 27 ½( 8 + 12) = ½ (8) + ½(12)  another matching problem ½( 20 ) = 4 + 6 10 = 10

11. How does this help? • Back to the original problem: • 5(3 - 4x) + 14x = 7(2 - 5x) • There are two parts of this problem that take the form a(b + c) • Can you see them?

12. We now have a path • We have a path forward. Let’s take it. • 5(3 - 4x) + 14x = 7(2 - 5x) becomes: • 5(3) + 5(-4x) + 14x = 7(2) + 7(-5x) • Notice here how we duplicate the 5 and the 7. • And we bring down anything else in the equation that was not affected, like the 14x here. • Also, we turn subtraction into the adding a negative number. This is so that the rule fits.

13. Now we carry out some math. • 5(3) + 5(-4x) + 14x = 7(2) + 7(-5x) • 15 + (-20x) + 14x = 14 + -35x • Now we can combine some like terms • 15 + (-20x) + 14x = 14 + -35x • 15 + (-6x) = 14 + -35x

14. Getting closer • 15 + (-6x) = 14 + -35x Now, to get this solved we need all of the x’s on one side and all of the numbers on the other. We also need x to be positive. What is the next step?

15. Move -35x to the left hand side • 12 + (-6x) = 14 + -35x • That’s right. -35x will be positive will become positive IF we move it to the other side. It belongs over there with the other x anyway. • Remember, like variables, like each other. • We should let them be together. • Numbers like numbers, and we don’t like them, “messing” with our variable, so we kick them to the other side.

16. Add to get rid of a negative Xs. • 15 + (-6x) = 14 + -35x • + 35x = + 35x • 15 + 29x = 14 + 0 • Next? • Get rid of the 15. That’s right. Subtract 15 from both sides.

17. Bye bye 15. • 15 + 29x = 14 • -15 = -15 • 29x = -1 • Almost there. Since 29 is still multiplying my x, I need to…. • Divide by 29, to get x alone.

18. Bye bye 29. Hello sweet solution!! • 29x = -1 • /29 = / 29 • x = -1/29 • DONE!! • We make a check mental check to see if that fraction is simplified. It is. So we have a solution. • BUT, with all that work there is a great chance that we made a mistake somewhere. So, plug your solution back in and check.

19. Plug and check • Go all the way back to the original equation. • 5(3 - 4x) + 14x = 7(2 - 5x) • Carefully put -1/29 in place of every x. • Many students for get to get rid of the x as they make a replacement. • 5(3 – 4(-1/29)) + 14(-1/29) = 7(2 - 5(-1/29)) • Xs gone, fractions in place. Let’s move on.

20. Be careful. • Making a mistake when you are checking will be just as bad as if you had made a mistake in the actual doing of the problem. • 5(3 – 4(-1/29)) + 14(-1/29) = 7(2 - 5(-1/29)) • 5(3 + 4/29) + ( -14/29 ) = 7(2 + 5/29 ) • Convert whole numbers to fractions with like denominators.

21. Moving right along. • 5(3 + 4/29) + ( -14/29 ) = 7(2 + 5/29 ) • 5(87/29 + 4/29) + (-14/29) = 7(58/29 +5/29) • 5(91/29) – (14/29) = 7(63/29) • 455/29 – 14/29 = 441 /29 • 441 / 29 = 441 / 29 The truth shall prove us right!!

22. Let’s review. • We got there, step by step. • We used the distributive property, in reverse. • We multiplied and added fractions. • We added the same thing to both sides. • We divided both sides by the same thing. • We kept our goals in mind and kept moving slowly towards them: • All copies of the variable must get to the same side. • The variable must be positive, on its side of the equals sign. • All numbers and operation must move to the OTHER side.

23. Stop here and practice • This is a good place to stop, but we have not completely covered the standard yet. • Next we will do a complex inequality.

24. Complex Equationslike 3(x-7) + 2(x+3)=90And InequalitiesPart 2 Alegbra 1 Standard 4

25. Are inequalities like equations? • Yes! • But there are important differences. • Inequalities, like equations, make a statement about the right side and the left side. • They say something. • And like equations. Sometimes they are saying something that is true, and sometimes they are saying something that is false.

26. What makes the difference? • Just like with equations, it is the value that we chose for our variable, that will make the difference between truth or falsity, but the similarity ends there. • Because…… • There is only ONE number is all of creation that will make an equations true, its solution. • But there are many, many numbers that will make and inequality true. In fact 1000 lifetimes would not • be enough to count them all.

27. Why the difference? • Why the difference? • Because the requirements to get into the solutions club, if you will, are much less restrictive, for an inequality, then they are for an equation. • Let’s look at an example.

28. Equation vs. Inequality • x = 7 vs. x > 7 • Numbers that make x = 7 true: • 7 • Numbers that make x > 7 true: • 7.1 8 21 312.3 399/6 50 million. • The list is endless.

29. See • However, please note that while there are a huge amount of numbers that make x > 7 true. It is not the case that ALL numbers make x > 7 true. • If I let x be 3, would this make a true statement? • 3 > 7 • Clearly not.

30. How do we get this under control? • We use a RULE, a list with dots, or sometimes a picture, to define and keep under control which numbers make an inequality true, and which numbers make it false. • This is called the “solution set” of an inequality.

31. Back to the standard • When we are asked to “solve” an inequality with have to create a very clear, and hard rule for what numbers make the statement true and what numbers make it false. • Like putting a fence at the border of a country, there should be no doubt as to who is in and who is OUT.

32. I HOPE this is review. Some of you may need this, so here it is: > means the left side is Greater than the right side. < means the left side is Lesser than the right side. ≥ means the left side is Greater than OR equal to the right side. ≤ means that the left side is Lesser than or equal to the right side.

33. A quick and cute memory trick. • The little line under the inequality sign is meant to represent the lower half of an equals sign. • So any time you see that you have an “or equal to” situation. • Think of the sign as a monster’s mouth. A monster is always hungry and will always pick the bigger meal. This is the Greater than monster. He wants the bigger meal on the left side. This is the Lesser than monster. He wants the bigger meal on the right side.

34. Back to work: • Consider: • -7y + 6 -3y < 2(y - 3) • We are trying to find replacements for y that make the left side LESS THAN the right side.

35. The rules are almost the same. • The goals and the rules are ALMOST exactly the same. • Isolate all copies of the variable to one side. • Move all numbers and operations that are not variables to the other side. • The variable must be made positive.

36. The unfortunate EXCEPTION • The are two huge changes to the normal rules of solving equations and they involve negative numbers. • If you multiply both sides by a negative number you must turn the inequality to point in the other direction. • If you divided by a negative number you must turn the inequality to point in the other direction. • For example: -3x < 9 /-3 /-3 Dividing both sides by -3 x > -3 Notice the sign must now FLIP to the other side. -1/9x > 2 * -9 * -9 Multiply both sides by -9. x < -18 Notice the sign must now FLIP to the other side.

37. Now, really back to work: -7y + 6 -3y < 2(y - 3) • We are trying to find replacements for y that make the left side LESS THAN the right side. • Apply the distributive property to the right side:

38. Distribute 2 on the right side. -7y + 6 -3y < 2(y - 3) -7y + 6 -3y < 2y – 6 Combine like terms: -10y + 6 < 2y -6 Get rid of the + 6 by subtracting 6. -6 -6 -10y < 2y -12 Why didn’t I FLIP? Because I am Subtracting here. The rule ONLY applies to multiplying or dividing, not to subtracting.

39. Moving right along -10y < 2y – 12 Subtract -2y from the right side. -2y -2y 12y < -12 Divide both sides by -12. NOW we flip. /-12 /-12 y > 1

40. Checking, checking, 1,2,3… • y > 1 This is our solution SET. So what we are claiming is that any replacement for y larger then 1 will make for a true statement in our original inequality. • Pop quiz: does this mean that 1 will solve our inequality?

41. We will pick 1.2 -7y + 6 -3y < 2(y - 3) -7(1.2) + 6 -3(1.2) < 2( 1.2 -3 ) replaced? -8.4 + 6 -3.6 < 2 ( -1.8) We chug some math. -6 < -3.6 This works. Keep in mind that being more negative makes -6 less than -3.6.

42. Let’s try 1… for fun. -7y + 6 -3y < 2(y - 3) -7(1) + 7 -3(1) < 2( 1 – 3) replaced right? -7 + 7 -3 < 2 (-2) - 3 < -4 This is a FALSE statement. So 1 does not work. And it should not. 1 is not less then 1, so it is not in the solution club, or the solution set.

43. Review: • Inequalities have a whole set of numbers that make them true, so a rule must be found, not just a number. • When you multiply or divide an inequality by a negative number, you must flip the inequality sign to point in the other direction. • REMEMBER, you do NOT apply this flipping rule for adding and subtracting, or for any other reason. • REMEMBER, all other rules for solving inequalities are identical to the rules for solving equations.

44. 5 -5 -4 -3 -2 -1 0 1 2 3 4 Displaying a solution set. • Use set brackets and a rule: { x | x > 1} reads the set of x such that x is less than 3. • Make a list and use … to say that the list continues. This is not precise for > or <: 1.0001, 1.0002, 1.0003, 1.0003 … • Draw a picture, using a number line or a number graph:

45. 5 -5 -4 -3 -2 -1 0 1 2 3 4 Important side note • As a general rule in math when we graph, we use ink, or a solid mark to represent truth. That is, if there is a mark there, then the statement is true there. If there is no mark there, or an OPEN mark there then it means that the statement if false, at that location. • For example, in this case we are representing the set: x > 1. The OPEN circle right on top of the one shows us that 1 cannot be exactly equal to one, but one is the border line. Right after one is the exactly point where numbers start to make the statement true, so any numbers to the right of one are true and this is represented by the solid red line.

46. 5 -5 -4 -3 -2 -1 0 1 2 3 4 All about the circles • So, if we want to graph and inequality on a number line. I start by placing a circle. The we fill that circle in if our inequality reads: “Greater than OR EQUAL TO” or if it reads “Lesser than OR EQUAL TO”. • Then we draw a line down the number line going in the same direction as our little arrow head is pointing. • > This little arrow head is pointing to the right. • < This little guy is pointing to the left. • Here we represent the set x ≥ -2:

47. 5 -5 -4 -3 -2 -1 0 1 2 3 4 5 -5 -4 -3 -2 -1 0 1 2 3 4 5 -5 -4 -3 -2 -1 0 1 2 3 4 x ≥ 2 Some more examples and done. x > 0 x ≤ 2

48. A few more. 5 -5 -4 -3 -2 -1 0 1 2 3 4 x < 4 5 -5 -4 -3 -2 -1 0 1 2 3 4 x ≤ -3 5 -5 -4 -3 -2 -1 0 1 2 3 4 x ≥ -4

49. Thank you. • Remember to practice. • Do your homework every night. • You will reach your goals. • This Power Point will be available at: • http://TeacherTube.com • http://WorldofTeaching.com • Visit our sponsor at: http://whaleboneir.com