1. Topic: EQUATIONS Simple Equations Fractional Equations

2. Guidelines Equations must be balanced. You must respect the laws of equations. The goals is to bring variable on the left and number to the right. Coefficient (or number) of variable +1. For example, 1x = 5; just put x = 5

3. Examples 1.#1) Solve: x = 5 + 2 x = 7 2.x = 5 (2 + 7) – ( 7 – 3) x = 10 + 35 – 7 + 3 Remember a minus before a bracket changes the sign of everything in the bracket x = 41

4. Guidelines con’t Whatever you do to one side, you must do to another. If you add 5 to one side, you must add 5 to the other side. If you have a number near the variable, always divide it by that number. For example 2x = 10; divide both by 2. x = 5 If -5x = 10; divide by -5; x = -2 When it changes signs, it changes signs

5. Example 4x + 1 = 13 4x = 13 - 1 (the 1 changed sides so it changes signs) 4x = 12 (divide both sides by 4) x = 3

6. Verification If you want to guarantee that you have the right answer you should verify. To verify: replace the number into the letter in the question. 4x + 1 = 13 (Original question) & x=3 4 (3) + 1 = 13 12+1=13 13=13 This is true; you have the right answer.

7. Guideline & Example 3x – 5 = 10x + 10 -3x - 10x = 10 + 5 -13x = 15 x = -1.15

8. Another Example – Long Version 3x – 5 = 8x + 15 3x – 8x – 5 + 5 = 8x – 8x + 15 + 5 -5x = 20 x = -4

9. Same example – Short Version 3x – 5 = 8x + 15 -5x = 20 x = -4 We will continue with the short version ;)

10. More examples 4x + 7 = 2x – 11 2x = -18 x = - 9 Verify! 4 (-9) + 7 = 2 (-9) – 11 -36 + 7 = -18 – 11 - 29 = -29 (You have the right answer)

11. Another Example 9x – 5 = 2x + 4 7x = 9 x = 1.29 You can still verify this! 9 (1.29) – 5 = 2 (1.29) + 4 6.61 = 6.58 (rounding error… close enough!)

12. More Examples 3x + 5 = 6x + 25 x = - 6.67 4x + 2 = 8x – 31 X = 8.25 --- are you verifying?)

13. Reminders Distributive property 5(3x + 2) means you multiply 5 by everything in the bracket 15x + 10 A minus sign before the bracket changes the sign of everything in the bracket

14. Now to add some fun – and have them longer 3 (5x-7) – (2x+8) = 2 (3x-1) 15x – 21 – 2x – 8 = 6x – 2 CLEAN IT UP BEFORE MOVING NUMBERS OR LETTERS 13x – 29 = 6x – 2 7x = 27 x = 3.86 Verify!

15. Verify 3 (5x-7) – (2x+8) = 2 (3x-1) 3 (5(3.86) – 7) – (2(3.86) - 8 = 2 (3(3.86) – 1) 3 (12.3) – 7.72 – 8 = 2 (10.58) 21.18 = 21.16 (rounding error – close enough!)

16. Quiz # 3 Equations 1) 5x – 7 = 3x + 7 2) 2x – 5 = 4x + 7 3) 9x – 2 = - 40 + 5x 4) 3x – 7 = 8x + 20 5) 6x – 1 = 8x + 20

17. Quiz #3 Con’t 6) 2 (3x – 7) = 5 (3x – 1) 7) 7(4x + 1) – 5 (3x + 5) = 8x – (3x + 2) 8) 2x – (3x + 1) 9) 2x + 5 10) – 5 – 5

18. Quiz #3 Equation Solutions 1) 7 (1a) 2) – 6 (1b) 3) -9.5 (1i) 4) – 5.4 (1o) 5) -10.5 6) -1 (1u) 7) 2 (1w) 8) –x – 1 (minus before a bracket!) 9) 2x + 5 (don’t mix apples & oranges!) 10) – 10

19. Fractional Equations Once again we want to get rid of the fraction Find the LCD (Lowest Common Denominator) Multiply every term to get the LCD.

20. Example 1. x – 2 = 11 5 3 15 LCD of 5, 3 and 15 is LCD is 15 3x – 10 = 11 3x = 21 X = 7 and then yes… VERIFY

21. Verify 1. x – 2 = 11 5 3 15 7/5 – 2/3 = 11 / 15 0.73 = 0.73 it works!

22. Another example 2x – 11 = 3x – 5 7 14 28 7 8x – 22 = 3x – 20 5x = 2 x = 0.4

23. Another Example 3 – 11x = 5 + 5x 8 12 24 6 9 – 22x = 5 + 20x -42x = -4 X = 0.1

24. Final Example (Hard) 3x – 2 – 11x + 8x = 3 – 7x + 1 5 3 30 15 2 15 30 18x – 20 – 11x + 16x = 45 – 14x +1 37x = 66 X = 1.78