1. Winter Warm up There are 25 students in my class. 17 said they would go snow skiing this year, 20 said they would go to Universal Studios and 4 would not do either. How many students will do both this semester? Snow skiing Universal Studios

2. Solution 16 4 4 Neither Snow skiing Universal Studios 1

3. Solving Inequalities Lesson 4-5 < < < > < > < <

4. An inequality is like an equation, but instead of an equal sign (=) it has one of these signs: < : less than ≤ : less than or equal to > : greater than ≥ : greater than or equal to

5. Review of Inequality Signs > greater than < less than greater than or equal less than or equal

6. How to graph the solutions > Graph any number greater than... open circle, line to the right < Graph any number less than... open circle, line to the left Graph any number greater than or equal to... closed circle, line to the right Graph any number less than or equal to... closed circle, line to the left

7. What do Inequalities mean? A mathematical sentence that uses one of the inequality symbols to state the relationship between two quantities.

8. Graphing Inequalities When we graph an inequality on a number line we use open and closed circles to represent the number. < < Plot a closed circle ≤ ≥ Plot an open circle

9. x < 5 means that whatever value x has, it must be less than 5. Try to name ten numbers that are less than 5!

10. Numbers less than 5 are to the left of 5 on the number line. 051015-20-15-10-5-252025 If you said 4, 3, 2, 1, 0, -1, -2, -3, etc., you are right. There are also numbers in between the integers, like 2.5, 1/2, -7.9, etc. The number 5 would not be a correct answer, though, because 5 is not less than 5.

11. x ≥ -2 means that whatever value x has, it must be greater than or equal to -2. Try to name ten numbers that are greater than or equal to -2

12. Numbers greater than -2 are to the right of -2 on the number line. 051015-20-15-10-5-252025 If you said -1, 0, 1, 2, 3, 4, 5, etc., you are right. There are also numbers in between the integers, like -1/2, 0.2, 3.1, 5.5, etc. The number -2 would also be a correct answer, because of the phrase, “or equal to”. -2

13. Solving an Inequality Follow the same rules and steps that we used to solve an equation. Always undo addition or subtraction first, then multiplication. Remember whatever is done to one side of the inequality must be done to the other side. The goal is to get the variable by itself.

14. Solve the inequality: x + 4 < 7 -4 -4 x < 3 ● Subtract 4 from each side. ● Keep the same inequality sign. ● Graph the solution. Open circle, line to the left. 051015-20-15-10-5-252025

15. Solve an Inequality w + 5 < 8 - 5 -5 w < 3 All numbers less than 3 are solutions to this problem! 051015-20-15-10-5-252025

16. More Examples 8 + r ≥ -2 -8 -8 r -10 All numbers greater than-10 (including -10) ≥ 051015-20-15-10-5-252025

17. More Examples 2x > -2 2 2 x > -1 All numbers greater than -1 make this problem true! 051015-20-15-10-5-252025

18. More Examples 2h + 8 ≤ 24 -8 -8 2h ≤ 16 2 2 h ≤ 8 All numbers less than 8 (including 8) 051015-20-15-10-5-252025

19. There is one special case. ● Sometimes you may have to reverse the direction of the inequality sign!! ● That only happens when you multiply or divide both sides of the inequality by a negative number.

20. Example: - 3y + 5 >23 -5 -5 -3y > 18 -3 -3 y < -6 ● Subtract 5 from each side. ● Divide each side by negative 3. ● Reverse the inequality sign. ● Graph the solution. Open circle, line to the left. 0-6

21. Your Turn…. Solve the inequality and graph the answer. 1.x + 3 > -4 2.6d > 24 3.2x - 8 < 14 4.-2c – 4 < 2 1.x > -7 2.d > 4 3.x < 11 4.c < -3

22. Try these: 5. 2x+3>x+5 6. - c - 11>23 7. 3(r-2)<2r+4 5.x > 2 6.c < 34 7.r < 10

23. Homework Complete the odds on the 5.1 and 5.2 workseets