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Mastering Linear Inequalities: Concepts and Practice

Learn how to solve and graph linear inequalities, including one-step, multi-step, and compound inequalities. Understand the symbols and rules involved in manipulating inequalities. Practice solving problems and interpreting graphs to master this essential algebraic concept.

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Mastering Linear Inequalities: Concepts and Practice

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  1. Algebra chapter 3 Solving and Graphing Linear Inequalities

  2. One-step linear inequalities—3.1

  3. Vocabulary • An equation is formed when an equal sign (=) is placed between two expressions creating a left and a right side of the equation • An equation that contains one or more variables is called an open sentence • When a variable in a single-variable equation is replaced by a number the resulting statement can be true or false • If the statement is true, the number is a solution of an equation • Substituting a number for a variable in an equation to see whether the resulting statement is true or false is called checking a possible solution

  4. Inequalities • Another type of open sentence is called an inequality. • An inequality is formed when and inequality sign is placed between two expressions • A solution to an inequality are numbers that produce a true statement when substituted for the variable in the inequality

  5. Inequality Symbols • Listed below are the 4 inequality symbols and their meaning < Less than ≤ Less than or equal to > Greater than ≥ Greater than or equal to Note: We will be working with inequalities throughout this course…and you are expected to know the difference between equalities and inequalities

  6. Graphs of linear inequalities • Graph (1 variable) • The set of points on a number line that represents all solutions of the inequality

  7. Graphs of linear inequalities

  8. Graphs of linear inequalities

  9. Writing linear inequalities • Bob hopes that his next math test grade will be higher than his current average. His first three test scores were 77, 83, and 86. • Why would an inequality be best in this case? • How can we come up with this inequality? • Graph! 

  10. Solving one-step linear inequalities • Equivalent Inequalities • Two or more inequalities with exactly the same solution • Manipulating Inequalities • All of the same rules apply to inequalities as equations* • When multiplying or dividing by a negative number, we have to switch the inequality! • Less than becomes greater than, etc.

  11. Solving with addition/subtraction

  12. Solving with addition/subtraction

  13. Solving with multiplication/division

  14. Solving with multiplication/division

  15. Why do we have to change the sign? • Is there another way we can solve this?

  16. Solving multi-step linear inequalities—3.2 Algebra chapter 3 • Solving and Graphing Linear Inequalities

  17. Multi step inequalities • Treat inequalities just like you would normal, everyday equations* *change the sign when multiplying or dividing by a negative!!

  18. Examples:

  19. Examples:

  20. Examples:

  21. Examples:

  22. Example • You plan to publish an online newsletter that reports the results of snow cross competitions. You do not want your monthly costs to exceed $2370. Your fixed monthly costs are $1200. You must also pay $130 per month to each article writer. How many writers can you afford to hire in a month?

  23. Examples: Try these on your own!

  24. o -5 -4 -3 o -5 -4 -3 ● -5 -4 -3 -5 -4 -3 Answer Now 1) Which graph represents the correct answer to > 1 ●

  25. Answer Now 2) When solving > -10will the inequality switch? • Yes! • No! • I still don’t know!

  26. Answer Now 3) When solving will the inequality switch? • Yes! • No! • I still don’t know!

  27. Answer Now 4) Solve -8p ≥ -96 • p ≥ 12 • p ≥ -12 • p ≤ 12 • p ≤ -12

  28. o -16 -15 -14 o -16 -15 -14 ● -16 -15 -14 -15 -15 -14 Answer Now 5) Solve 7v < -105 ●

  29. Class work:p.343 #15-37 oddIf you do not finish in class, then it becomes homework!

  30. Compound inequalities—3.6 Algebra chapter 3 • Solving and Graphing Linear Inequalities

  31. Compound inequality • What does compound mean? • Compound fracture? • So…what’s a compound inequality? • An inequality consisting of two inequalities connected by an and or an or

  32. Graphing Compound Inequalities • Graph the following:

  33. Graphing Compound Inequalities • Graph the following:

  34. Graphing Compound Inequalities • Graph the following: • All real numbers that are greater than or equal to -2 and less than 3

  35. Solving Compound inequalities • Again….treat these like equations! • Whenever we do something to one side… …We do it to every side!

  36. Solving Compound Inequalities

  37. Solving Compound Inequalities

  38. Solving Compound Inequalities

  39. Solving Compound Inequalities

  40. homework:p.349 #12-36 even

  41. Solving Absolute-Value Equations and Inequalities—3.6 (Day 1)

  42. Abs. Value • What is Absolute Value? • Distance from zero • What does that mean?

  43. Abs. Value • So….an absolute value equation has how many solutions? • Is this always true?

  44. Abs. Value • How do we apply this to equations? • Ex:

  45. Examples

  46. Examples

  47. Examples

  48. Examples

  49. Examples

  50. p.356#19-36

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