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1.4 Absolute Values

Understand absolute value equations, inequalities, compound values, and exponential rules in math. Learn about radicals, rational exponents, and simplifying expressions. Practice problems and techniques step by step.

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1.4 Absolute Values

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  1. 1.4 Absolute Values Solving Absolute Value Equations By putting into one of 3 categories

  2. What is the definition of “Absolute Value”? __________________________________ __________________________________ Mathematically, ___________________ For example, in , where could x be? 0 3 -3

  3. To solve these situations, ______________________________________________________________________________________________________ Consider ________________________ 0

  4. That was Category 1 Category 1 : ____________________________ _______________________________________ _______________________________________ Example Case 1 Case 2

  5. Absolute Value Inequalities Think logically about another situation. What does mean? For instance, in the equation , _________________________________ 0

  6. How does that translate into a sentence? __________________________________ Now solve for x. This is Category 2: when x is less than a number

  7. Absolute Value Inequalities What does mean? In the equation , __________________________________ 0

  8. Less than = And statement Greater than = Or statement Note:  is the same as ; is the same as ; just have the sign in the rewritten equation match the original.

  9. Isolate Absolute Value • _______________________________________ • ______________________________________ • ______________________________________

  10. When x is on both sides • ______________________________________ ________________________________________ • ______________________________________ Case 1 Case 2

  11. Example inequality with x on other side Case 1 Case 2

  12. Examples Case 1 Case 2

  13. Case 1 Case 2

  14. Case 1 Case 2

  15. _________________________________ Whats wrong with this? ____________________________________________________________________________________

  16. 1-4 Compound Absolute ValuesEqualities and Inequalities More than one absolute value in the equation

  17. Some Vocab • Domain- _________________________ • Range- __________________________ • Restriction- _______________________ • __________________________________________________________________

  18. Find a number that works. We will find a more methodical approach to find all the solutions.

  19. In your approach, think about the values of this particular mathematical statement in the 3 different areas on a number line. • -_________________________________________ • ________________________________________ • - ________________________________________

  20. It now forms 3 different areas or cases on the number line. • ______________________________ • ______________________________ • ___________________________________

  21. Case 1 Domain ________________________ ____________________________________________________ ___________________________________________________________________________________________________

  22. Case 2 Domain ___________________________________

  23. Case 3 Domain _____________________________

  24. Final Answers • ______________________________ • ______________________________ • _____________________________ 2 5 -2

  25. If you get an answer in any of the cases where the variable disappears and the answer is TRUE ________________________________ ________________________________________________________________________________________________

  26. 1-5 Exponential Rules You know these already

  27. Review of the Basics

  28. Practice problems Simplify each expression

  29. Do NOT use calculators on the homework, please!! 

  30. Rule:If bases the same set exponents equal to each other

  31. 1-6 Radicals (Day 1) and Rational Exponents (Day 2)

  32. What is a Radical? In simplest term, it is a square root ( ) = ________________

  33. The Principle nth root of a: 1. • If a < 0 and n is odd then is a negative number ___________________ • If a < 0 and n is even, _____________ __________

  34. Vocab

  35. Lets Recall

  36. Rules of Radicals

  37. Practice problems Simplify each expression

  38. Class Work problems Simplify each expression

  39. 1-6 Radicals (Day 1) and Rational Exponents (Day 2)

  40. What is a Rational Exponent? ____________________________________ ____________________________________ ____________________________________ ____________________________________

  41. Don’t be overwhelmed by fractions! These problems are not hard, as long as you remember what each letter means. Notice I used “p” as the numerator and “r” as the denominator. p= _____________________________ r= _____________________________ ________________________________________ ________________________________________ ________________________________________

  42. Practice problems Simplify each expression

  43. The last part of this topic What is wrong with this number? ___________________________________ ___________________________________ ___________________________________

  44. Rationalizing If you see a single radical in the denominator ______________________________________ ________________________________________ ________________________________________ ________________________________________

  45. Rationalizing If you see 1 or more radicals in a binomial, what can we do?

  46. What is a Conjugate? The conjugate of a + b ______. Why? The conjugate _______________________ ___________________________________ ___________________________________ ___________________________________

  47. Rationalizing Multiply top and bottom by the conjugate! Its MAGIC!!

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