1. Solving Equations 08/09/12lntaylor ©

2. Table of Contents Learning Objectives Evaluating Expressions Solving Equations Solving Inequalities Practice 1 2 3 4 08/09/12lntaylor © 5

3. Learning Objectives LO 1Understand the difference between evaluating and solving TOC LO 2Correctly evaluate expressions LO 3Correctly solve equations and inequalities 08/09/12lntaylor ©

4. Definitions Definition 1Evaluate – “Plug in” a value(s) and simplify an expression/equation TOC Definition 4Equations – contain = Definition 5Inequalities – contain ≠ ≤ ≥ Examples 08/09/12lntaylor © Expression -> 2x + 3 Equation -> y = 2x + 3 Inequality -> y > 2x + 3 Definition 3Expressions – do not contain = ≠ ≤ ≥ Definition 2Solve – Using reverse operations, find value(s) that fit an equation

5. Previous Knowledge PK 1Basic Operations and Properties PK 2Simplify by Combining Like Terms TOC 08/09/12lntaylor © PK 3Exponent Rules

6. Evaluating Expressions TOC 08/09/12lntaylor ©

7. Step1Evaluate means you need two things An expression or equation Some value TOC 08/09/12lntaylor © Step2Evaluate means “Plug it in!” Step3Simplify Evaluate 2x + 3 when x = 42x + 3x = 4 2 (4) + 3 = 8 + 3 = 11 2x + 3 =

8. Now you try! Evaluate 5x + 6 when x = − 4 TOC 08/09/12lntaylor ©

9. Step1Evaluate means you need two things An expression or equation Some value TOC 08/09/12lntaylor © Step2Evaluate means “Plug it in!” Step3Simplify Evaluate 5x + 6 when x = − 4 5x + 6 x = − 4 5 (− 4) + 6 = − 20 + 6 = − 14 5x + 6 =

10. Now you try something harder! Evaluate 5x 2 + 6x + 10 when x = 2 TOC 08/09/12lntaylor ©

11. Step1Evaluate means you need two things An expression or equation Some value TOC 08/09/12lntaylor © Step2Evaluate means “Plug it in!” Step3Simplify Evaluate 5x 2 + 6x + 10 when x = 2 5x 2 + 6x + 10x = 2 5 (2 2 ) + 6(2) + 10 = 5(4) + 12 + 10 = 5x 2 + 6x + 10 = 20 + 12 + 10 = 42

12. Now you try a really hard one! Evaluate 4x 2 + 5x + 22 x 3 when x = - 2 TOC 08/09/12lntaylor ©

13. Step1Evaluate means you need two things An expression or equation Some value TOC 08/09/12lntaylor © Step2Evaluate means “Plug it in!” Step3Simplify Evaluate 4x 2 + 5x + 18 / x 3 when x = – 2 4x 2 + 5x + 18 / x 3 x = – 2 4(– 2 2 ) + 5(– 2) + 18 / (– 2) 3 = [4(4) – 10 + 18] / – 8 = 4x 2 + 5x + 18 / x 3 = [16 – 10 + 18]/ – 8 = 24/ – 8 = – 3 NoteThe only reason the problem is “hard” is it has a – sign! Always scan the problem for – signs before starting a problem

14. Solving Equations TOC 08/09/12lntaylor ©

15. Step1Solve means LOL LOL means Letter on Left Everything else goes to the right by using reverse operations! TOC 08/09/12lntaylor © Step2I am going to use a teeter totter to demonstrate Rewrite the equation (= over the fulcrum) The = means the equation is in “balance” at all times! NoteI will show you what happens when you do not do the opposite! Move any number not attached to a variable(letter). Here I will move the 6. Watch what happens! Solve 5x + 6 = 46 5x + 6 = 465x = 46+ 6

16. Step3Let’s do it the correct way LOL means Letter on Left Everything else goes to the right by using reverse operations! TOC 08/09/12lntaylor © Step4What is the opposite operation of + 6? Note that the equation is still in balance Step5What is the opposite operation of 5x? Note that the operation is still in balance Solve 5x + 6 = 46 5x + 6 = 465x = 46− 6x= 40 / 5 Step6The solution is… Until you get really good, check by “Plugging it in”! x = 8 5(8) + 6 = 46 40 + 6 = 46 46 = 46

17. Now you try Solve 4x − 10 = 90 TOC 08/09/12lntaylor ©

18. Step1Solve means LOL LOL means Letter on Left Everything else goes to the right by using reverse operations! TOC 08/09/12lntaylor © Step2Use a teeter totter for now Rewrite the equation (= over the fulcrum) The = means the equation is in “balance” at all times! Step3Move any number not attached to a variable(letter) Use the opposite operation Continue until you solve the problem! Solve 4x – 10 = 90 4x – 10 = 90 4x = 90 + 10 Step4Plug in your answer and check your work! 4x = 100 / 4 x = 100 x = 25 4(25) – 10 = 90 100 – 10 = 90 90 = 90

19. Now you try without the teeter totter! Solve 2(4x − 3) = 3x + 15 TOC 08/09/12lntaylor ©

20. Step1Solve means LOL LOL means Letter on Left Everything else goes to the right by using reverse operations! TOC 08/09/12lntaylor © Step2No need for the teeter totter! The = means the equation is in “balance” at all times! Step3Distribute Move like terms together (LOL) Use the opposite operation Continue until you solve the problem! Solve 2(4x – 3) = x + 15 Step4Check your answer 8x – 6 = x + 15 8x = x + 15 + 6 8x = x + 21 8x – x = 21 7x = 21 x = 21/7 x = 3 2[(4(3) – 3] = 3 + 15 2[12 – 3] = 3 + 15 2[9] = 3 + 15 18 = 3 + 15 18 = 18 + 6 – x /7

21. Now you try! Solve (14x − 3)/5 = 4x + 15 TOC 08/09/12lntaylor ©

22. Step1Solve means LOL LOL means Letter on Left Everything else goes to the right by using reverse operations! TOC 08/09/12lntaylor © Step2No need for the teeter totter! The = means the equation is in “balance” at all times! Step3Distribute if/when needed Move like terms together (LOL) Use the opposite operation Continue until you solve the problem! Solve (14x – 3)/5 = 4x + 15 Step4Check your answer 14x – 3 = 5(4x + 15) 14x – 3 = 20x + 75 14x = 20x + 75 + 3 14x = 20x + 78 14x – 20x = 78 – 6x = 78 x = – 13 [14(-13) – 3]/5 = 4(-13) + 15 [– 182 – 3]/5 = – 52 + 15 – 185 / 5 = – 52 + 15 – 37 = – 52 + 15 + 3 – 20x / – 6 5( ) x = 78/ – 6 – 37 = – 37

23. Solving Inequalities TOC 08/09/12lntaylor ©

24. Step1Inequalities are solved like equations LOL means Letter on Left Everything else goes to the right by using reverse operations! TOC 08/09/12lntaylor © Step2The only difference is when you divide or multiply by a negative Flip the symbol around! Step3Distribute if/when needed Move like terms together (LOL) Use the opposite operation Continue until you solve the problem! Solve [14x – 3]/5 ≤ 4x + 15 Step4Check your answer Choose an easy number to work with; here 0 > -13 so we will use 0 14x – 3 ≤ 5[4x + 15] 14x – 3 ≤ 5 ∗ 4x + 5 ∗ 15 14x – 3 ≤ 20x + 75 14x ≤ 20x + 75 + 3 14x – 20x ≤ 78 – 6x ≤ 78 x ≥ – 13 [14(0) – 3]/5 ≤ 4(0)+ 15 (– 3)/5 ≤ 15 – 0.6 ≤ 15 This is True The answer is correct x ≥ 78/– 6

25. Now you try! Solve − 2[3x − 4] ≥ 4x − 12 TOC 08/09/12lntaylor ©

26. Step1Inequalities are solved like equations LOL means Letter on Left Everything else goes to the right by using reverse operations! TOC 08/09/12lntaylor © Step2The only difference is when you divide or multiply by a negative Flip the symbol around! Step3Distribute if/when needed Move like terms together (LOL) Use the opposite operation Continue until you solve the problem! Solve – 2 [3x – 4] ≥ 4x – 12 Step4Check your answer Choose an easy number to work with; here 0 < 2 so we will use 0 – 2 ∗ 3x – 2(– 4) ≥ 4x – 12 – 6x + 8 ≥ 4x – 12 – 6x ≥ 4x – 12 – 8 – 6x ≥ 4x – 20 – 6x – 4x ≥ – 20 – 10x ≥ – 20 x ≤ 2 – 2[3(0) – 4] ≥ 4(0) – 12 – 2(– 4) ≥ – 12 8 ≥ - 12 This is True The answer is correct x ≤ – 20 /– 10

27. Now you try one last one! Solve 2[4x − 5] ≥ 4x − 12 TOC 08/09/12lntaylor ©

28. Step1Inequalities are solved like equations LOL means Letter on Left Everything else goes to the right by using reverse operations! TOC 08/09/12lntaylor © Step2The only difference is when you divide or multiply by a negative Flip the symbol around! Step3Distribute if/when needed Move like terms together (LOL) Use the opposite operation Continue until you solve the problem! Solve 2 [4x – 5] ≥ 4x – 12 Step4Check your answer Choose an easy number to work with; here 0 ≥ - 1/2 so we will use 0 2 ∗ 4x + 2(– 5) ≥ 4x – 12 8x – 10 ≥ 4x – 12 8x ≥ 4x – 12 + 10 8x ≥ 4x – 2 8x – 4x ≥ – 2 4x ≥ – 2 x ≥ - ½ 2[4(0) – 5] ≥ 4(0) – 12 2(– 5) ≥ – 12 – 10 ≥ – 12 This is True The answer is correct x ≥ – 2/4

29. Practice TOC 08/09/12lntaylor ©

30. 08/09/12lntaylor © TOC ProblemAnswer Evaluate 3x – 7 when x = 8 Evaluate 3x 2 when x = 5 Evaluate – 3x 2 when x = -10 Evaluate – 4x 2 + 5x – 10 when x = 5 Solve 4x – 10 = 90 Solve 2(x – 10) = 80 Solve 4(3x + 30)/5 = 120 Solve 4x + 3 ≤ 31 Solve – 2(3x – 15) > – 60 > 17 > > > > > 75 – 300 – 85 x = 25 x = 50 > x = 40 > x ≤ 7 > x < 15 clear answers