1. MTH 070 Elementary Algebra Chapter 2 Equations and Inequalities in One Variable with Applications 2.4 – Linear Inequalities Copyright © 2010 by Ron Wallace, all rights reserved.

2. Inequality A statement that one expressions is …, or ≥, or ≥ … a second expression. Applications involving inequalities involve terms such as “at least” (≥) and “at most” (≤).

3. Solving an Inequality Determine ALL values of the variable that makes the inequality a true statement. Try some … 0? 1? -3? 10? 7? 5? 20? 8?

4. Solutions of Inequalities Three possible forms … Three possible forms … Simple inequality Simple inequality x  2 x  2 x < –3 x < –3 Interval notation Interval notation [2,  ) [2,  ) (– , –3) (– , –3) Graph Graph

5. Review – Solving Equations Eliminate grouping symbols. Eliminate grouping symbols. Combine like terms. Combine like terms. Eliminate terms Eliminate terms Add opposites to other side Add opposites to other side i.e. addition property i.e. addition property Eliminate factors Eliminate factors Multiply by reciprocals on the other side Multiply by reciprocals on the other side i.e. multiplication property i.e. multiplication property What happens when you do these things to inequalities? NO PROBLEM !

6. The Addition Property w/ Inequalities What happens if you add the same amount of weight to both sides of the scale?

7. The Addition Property w/ Inequalities The inequality relationship remains the same. Works with subtraction too!

8. The Addition Property w/ Inequalities If an expression is added to (subtracted from) both sides of an inequality, the result will be an equivalent inequality (i.e. same solutions).

9. The Multiplication Property w/ Inequalities What happens if you multiply each weight by the same POSITIVE value?

10. The Multiplication Property w/ Inequalities The inequality relationship remains the same. Works with division too! NOTE: C > 0

11. The Multiplication Property w/ Inequalities What happens if you multiply each weight by the same NEGATIVE value?

12. The Multiplication Property w/ Inequalities The inequality relationship reverses its direction. Works with division too! NOTE: C < 0

13. The Multiplication Property w/ Inequalities If both sides of an inequality are multiplied or divided by a positive expression, the result will be an equivalent inequality (i.e. same solutions). If both sides of an inequality are multiplied or divided by a positive expression, the result will be an equivalent inequality (i.e. same solutions). If both sides of an inequality are multiplied or divided by a negative expression AND the direction of the inequality is reversed, the result will be an equivalent inequality (i.e. same solutions). If both sides of an inequality are multiplied or divided by a negative expression AND the direction of the inequality is reversed, the result will be an equivalent inequality (i.e. same solutions).

14. Switching Sides If a < b, then how is b related to a? If a < b, then how is b related to a? b > a If a > b, then how is b related to a? If a > b, then how is b related to a? b < a Likewise for ≤ and ≥.

15. Solving Inequalities – Strategy (just like equations w/ one exception) Eliminate grouping symbols Eliminate grouping symbols Combine like terms Combine like terms Addition principle for inequalities Addition principle for inequalities Multiplication principle for inequalities Multiplication principle for inequalities Careful w/ this one! Careful w/ this one! If the variable is on the right; switch sides If the variable is on the right; switch sides Don’t forget to reverse the inequality symbol. Don’t forget to reverse the inequality symbol. Basic Goal: x > ? or x < ? or x ≥ ? or x ≤ ?

16. Checking Solutions Need to check 3 values … (okay, maybe 2 will do) Need to check 3 values … (okay, maybe 2 will do) Assume the solution: x < 3 Assume the solution: x < 3 Check x = 3 … this should make both sides equal Check x = 3 … this should make both sides equal Check any value less than 3 … this should make the original inequality TRUE. Check any value less than 3 … this should make the original inequality TRUE. Check any value greater than 3 … this should make the original inequality FALSE Check any value greater than 3 … this should make the original inequality FALSE Either of these will do. Hint: When checking inequalities, always check the number 0.

17. Solving & Checking Inequalities Example 1 of 5 Give solutions in all three forms

18. Solving & Checking Inequalities Example 2 of 5 Give solutions in all three forms

19. Solving & Checking Inequalities Example 3 of 5 Give solutions in all three forms

20. Solving & Checking Inequalities Example 4 of 5 Give solutions in all three forms

21. Solving & Checking Inequalities Example 5 of 5 Give solutions in all three forms