Solving Linear Equations and Inequalities

1. Solving Linear Equations and Inequalities B. Myers Unit 2

2. Solving Equations Using Addition & Subtraction We can solve equations by using transformations to isolate the variable on one side of the equation. When we rewrite equations using transformations we are writing equivalent equations

3. Solving Equations Using Addition & Subtraction To change or transform an equation into an equivalent equation think of a balance (what you do to one side of the equal sign, you have to do to the other) X + 3 = 5 & X = 2 are equivalent…how???

4. Inverse Operations Inverse Operations – operations that undo each other addition and subtraction multiplication and division These operations help us isolate the variable

5. Interchange The sides of an equation can be interchanged 10 = X can also be written as X = 10

6. Solution Step Each time you apply a transformation to an equation, solution steps are written one below the other with the = lined up.

7. Transformations That Produce Equivalent Equations Add the same number to both sides. Looks like: A Subtraction problem

8. Transformations That Produce Equivalent Equations Subtract the same number from each side. Looks like: An Addition problem

9. Transformations That Produce Equivalent Equations Simplify one or both sides (combine like terms)

10. Real-Life Problem Your bank balance is $42. If you write a check to buy a pair of shoes, your balance would be -$5. How much do the shoes cost? Write an equation to model the situation.

11. Solving Equations Using Multiplication and Division We still want to isolate the variable on one side of the equation. Remember that multiplication and division are inverse operations (so they undo each other)

12. Transformations That Produce Equivalent Equations Multiply each side of the equation by the same nonzero number. Looks like: Division Problem

13. Transformations That Produce Equivalent Equations Divide each side of the equation by the same nonzero number Looks like: Multiplication problem

14. Solving Multi-Step Equations Now we will be using 2 or more transformations to solve an equation

15. Solving Equations with Variables on Both Sides Collect variable terms on the side of the greater variable coefficient. (the one that has the bigger number with the variable)

16. Identity Some equations are ‘identity equations’ – that means the equation is true for all values of the variable You will know an equation is an identity when you get to the final solution step and the same number or variable on both sides of the = **Some equations have NO SOLUTIONS

17. Solving Decimal Equations In real-life exact solutions are not always practical. Sometimes we need to use rounding which can lead to round-off error.

18. Solving Inequalities • Solving one-step inequalities using addition or subtraction works the same way as solving equations using addition or subtraction! • Just solve the same way you would if the < or > was an =

19. Solving Inequalities • Solving inequalities using multiplication and division is a little different than solving equations using multiplication and division. • Here’s the difference: when you multiply or divide by a NEGATIVE number, you have to REVERSE the sign.