1. BY: KAYLEE J. KAMRYN P. CLOE B. EXPRESSIONS * EQUATIONS * FUNCTIONS * AND INEQUALITIES

2. EXPRESSIONS What is a math expression? Numbers, symbols and operators (such as + and ×) grouped together that show the value of something. Example: 2×3 is an expression

4. EXAMPLE Write an expression for the problem. The price of a visit to the dentist is $50. If the dentist fills any cavities, an additional charge of $100 per cavity gets added to the bill. If the dentist finds ( n ) cavities, what will the cost of the visit be? Write your answer as an expression. $50 + $100n

5. EXAMPLE Write an expression for the following: Eight less than 5 times a number Answer: _________

6. Solution Eight less than 5 times a number Answer: 5x - 8

7. EQUATIONS x + 8 – 3 = 15 (z) x 3 + 5 = 29

8. EQUATIONS An equation says that two things are equal. It will have an equal sign "=" like this: x+2=6 That equation says: what is on the left (x + 2) is equal to what is on the right (6) So an equation is like a statement "this equals that"

9. Different parts of an Equation: A Variable is a symbol for a number we don't know yet. It is usually a letter like x or y. A number on its own is called a Constant. A Coefficient is a number used to multiply a variable (4x means 4 times x, so 4 is a coefficient) A Operator is a symbol (such as +, x, -, etc.) that shows an operation or what we want to do with the values.

10. 1- step Equation Solve 3x = 9 1.To undo the multiplication, divide both sides by 3. 3/3 = 9/3 x = 3 Check: Replace x with 3 in the above equation. 3(3) = 9, 9 = 9

11. 2-Step Equation 5x + 8 = 43 Subtract 8 from both sides to undo the addition. 5x + 8 – 8 = 43 – 8 5x = 35 Divide both sides by 5 to undo the multiplication 5x/5 = 35/5 X = 7

12. Writing an equation for a function Neil writes 2 pages per hour. Write an equation that shows the relationship between the hours spent writing x and the total pages written y. Answer: y = 2x

13. Practice For the equation x = 3y − z, what is the value of x when y = 4 and z = 1? Solve the Equation : 5p = -30

14. Solution 1. For the equation x = 3y − z, what is the value of x when y = 4 and z = 1? x = 3(4) – 1 x = 12 – 1 x = 11 2. Solve : 5p = -30 Divide both sides by 5 to undo the multiplication: 5p/5 = -30/5 Answer: p = -6 Check: 5(-6) = -30

15. FUNCTIONS x698111518 y25471114 Y = x - 4

16. FUNCTIONS A function is a rule that relates 2 quantities so that each input value corresponds exactly to one output value. There are 3 main parts: The Input The Relationship (the pattern or function rule) The Output

17. Writing Equations from Function Tables x3456710 y811__ 1.Compare x and y to find a pattern 2.Make sure your pattern works for all the numbers in the table. * Well, I know that 3 times 3 is 9 and if I subtract 1 it will give me 8. So lets see if it works for the 2 nd input: 4 x3 = 12 – 1 = 11. So yes it does, this is the pattern or the “function rule” 3. Use the pattern to rewrite an equation. y = 3x – 1 NOW YOU TRY! What is the output for the rest of the numbers?

18. Solution x3456710 y81114172029 Y = 3x -1 HOW DID YOU DO?

19. Example : y = 2x Here’s a table where you can see the relationship or function rule Input x RelationshipOutput y 00x20 11x22 77x214 1010x220 5050x2? For an input of 50, what is the output?

20. Translating Words to Math Write an equation and solve: 1. Eight less than 6 times a number is sixteen. Answer: 6x – 8 = 16 X= 4 ***Always read the directions so you know what you need to do in the problem and turn it around when you see “than”***

21. Special Rules of a Function ** The same rule must work for every possible input value. An input and its matching output are together called an ordered pair So a function can also be seen as a set of ordered pairs

22. INEQUALITIES X > 1 Y < 3 X + 3 > 2 2y + 1 < 7

23. Inequalities Inequality tells us about the relative size of two values. Mathematics is not always about "equals"! Sometimes we only know that something is bigger or smaller

24. Example: Alex and Billy have a race, and Billy wins! What do we know? We don't know how fast they ran, but we do know that Billy was faster than Alex: Billy was faster than Alex We can write that down like this: b > a (Where "b" means how fast Billy was, ">" means "greater than", and "a" means how fast Alex was) We call things like that inequalities (because they are not "equal")

25. Greater than or Less Than The 2 most common inequalities are “ > ” – greater than Example: 5 > 2 “< “ – less than Example: 7 < 9 We can also have inequalities that include “equals” “ > “ – greater than or equal to “ < “ – less than or equal to **Helpful reminder: The open mouth part “eats” the bigger number, or the point part is always pointing at the smaller number.

26. What affects the direction of the Inequality * These are things we can do without affecting the direction of the inequality: Add (or subtract) a number from both sides Multiply (or divide) both sides by a positive number Simplify a side *But these things will change the direction of the inequality (" " for example): Multiply (or divide) both sides by a negative number Swapping left and right hand sides

27. Affecting the Inequality Example: 3x < 7+3 We can simplify 7+3 without affecting the inequality: 3x < 10 ________________________________ Example: 2y+7 < 12 When we swap the left and right hand sides, we must also change the direction of the inequality: 12 > 2y+7

28. Adding or Subtracting a Value Solve: x + 3 < 7 If we subtract 3 from both sides to undo the addition, we get: x + 3 - 3 < 7 - 3 x < 4 And that is our solution: x < 4 In other words, x can be any value less than 4. What did we do? We went from this: To this:

29. Practice Solve the Inequality q – 12 > 3 87 < 25 + x

30. Solution 1.q – 12 > 3 Add 12 to both sides to undo the subtraction q – 12 + 12 > 3 + 12 Q > 15 2.87 < 25 + x Subtract 25 from both sides to undo the addition 87-25 < 25 –25 + x 62 < x

31. Graphing an Inequality 1.Make a number line that includes negative numbers on the left and positive numbers on the right with a 0 in the middle separating the negative from the positive. 2Identify any circles that need to be put on the number line. If you have a less than or greater than sign, you will put an open circle. If you have a less than or greater than sign AND an equal sign “ ” then you will put a closed circle. 3Draw your arrow on the number line that follows your inequality. If it’s a less than sign, draw your arrow to the left, if its greater than, draw your arrow to the right.