1. SECTION 1-4: Solving Inequalities We solve inequalities the same way we solve equations with the following exception: **GOLDEN RULE for Inequalities** When you multiply or divide by a __________ number, you MUST _______ the direction of the inequality symbol.

2. The Inequality Symbols Key words that describe each symbol: 1. < - less than, 2. > - greater than, 3. ≤ - less than or equal to, 4. ≥ - greater than or equal to,

3. Solving Inequalities - EXAMPLES EX.A) -3x > 6 B) 2 - x ≤ -7

4. Graphing Solutions of Inequalities Rules for graphing inequalities: - use an ________ dot ≤ or ≥ - use a _________ dot < or ≤ - shade to the ________ > or ≥ - shade to the ________ ** The variable must be on the _______ after you solve to use these rules!! (Ex. x < 3)

5. Graph the solution: EXAMPLES EX.A) 3x – 12 < 3 Graph:  ------------------------------------  Is ___ part of the solution?

6. Check your answer How can we check our answer to EX.A if 5 is not part of the solution??

7. EXAMPLES – Graphing the Solution EX.B) 9 – 2x > 5 EX.C) 3x – 7 ≤ 5  ---------------------- 

8. ALL REAL NUMBERS & NO SOLUTION When our result has no variable left in it, our answer is either all real numbers or no solution. If the result is _______ (Ex. 3 < 7), our answer is ________________________________. If the result is _______ (Ex. 3 > 7), our answer is ________________________________.

9. EXAMPLES EX.1) 2x – 3 > 2(x – 5) Our result is ______. Therefore, our answer is ___________________________. Graph:  ---------------------------------- 

10. EXAMPLES EX.2) 7x + 6 < 7(x – 4) Our result is ______, therefore our answer is _______________________. Graph:  -------------------------------- 

11. EXAMPLES – Try These: 1) 2x < 2(x + 1) + 3 2) 4(x – 3) + 7 ≥ 4x + 1 3) 4x + 8 > -4(x – 8)

12. INEQUALITY WORD PROBLEMS - write an inequality for the situation EX. A band agrees to play for $200 plus 25% of the ticket sales. Find the ticket sales needed for the band to receive at least $500. Define variables: Let x = __________________ In words, $200 + 25% ticket sales _______ $500 Write an inequality:

13. Inequality word problems… Solve the inequality: Write a sentence for your answer: _________ _______________________________________

14. Inequality word problems…Example 2 A salesperson earns a salary of $700 per month plus 2% of the sales. What must the sales be if the salesperson is to have monthly income of at least $1800. Let x = _____________________________ Write an equation:

15. Example 2, continued… Solve the inequality: Write a sentence for your answer: _________ _______________________________________

16. Example 3 The lengths of the sides of a triangle are 3:4:5. What is the length of the longest side if the perimeter is not more than 84 cm? Use x to represent the ratio. s 1 = s 2 = s 3 =

17. Example 3, continued… Write an inequality from the given information: What is the length of the longest side??

18. COMPOUND INEQUALITIES Compound inequalities are ________ of inequalities joined by _______ or ________. If ‘and’ and ‘or’ are not written, use the following rule: Less thAN(<, ≤)  use ANd GreatOR (>, ≥)  use OR

19. ‘AND’ Graphs AND represents the overlap, also called the ___________ of the two inequalities. We need to transfer everything with 2 lines above onto our final graph. EX.  ----------------------------------- 

20. ‘AND’ Examples 3x – 1 > -28 AND 2x + 7 < 19 STEP 1: Solve each inequality separately Step 2: Graph each above the final number line Step 3:  ---------------------------------- 

21. ‘AND’ Examples 2x < x + 6 < 4x – 18 (less thAN  use AND) STEP 1: Solve each inequality separately Step 2: Graph each above the final number line Step 3:  ---------------------------------- 

22. ‘OR’ Graphs OR represents the ________ of the two inequalities. We need to transfer everything with 1 or more lines above onto our final graph. EX.  ----------------------------------- 

23. ‘OR’ Examples 4y – 2 ≥ 14 OR 3y – 4 ≤ -13 STEP 1: Solve each inequality separately Step 2: Graph each above the final number line Step 3:  ---------------------------------- 

24. ‘OR’ Examples x - 12 ≥ -5x ≥ -2x – 9 (greatOR  use OR) STEP 1: Solve each inequality separately Step 2: Graph each above the final number line Step 3:  ---------------------------------- 