Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20R.1 Real Numbers Chapter R Section 2: Algebra EssentialsIn this section, we will… Complete and graph inequalities Find distance on the real number line; compute absolute values Evaluate algebraic expressions and equations for given values Determine the domain of a variable Use the Laws of Exponents to evaluate expressions Evaluate square rootsR.2 Algebra Essentials: Complete and Graph InequalitiesThe Real Number Line: negative real numbers positive real numbers zero a b b a a bR.2 Algebra Essentials: Complete and Graph Inequalities532- -2 1.4120.6663Examples: Make each of the following statements true using < , > or = .Examples: On the real number line, graph all numbers x for which4x >2x �- is negativexR.2 Algebra Essentials: Find Distance on the Real Number Line; Compute Absolute ValuesThe absolute value of a real number a, denotedis the distance from a to zero.Examples: Evaluate each of the following43.7- -0a0 if 0a a a= � if 0a a a=- <3.7-R.2 Algebra Essentials: Find Distance on the Real Number Line; Compute Absolute ValuesExample: Evaluate the following for a = 5 and b = -3a b+a b+b a-a b-Note: Do not guess…be sure to always follow the order of operations!Note: Do not guess…be sure to always follow the order of operations!Note: The distance from a to b is the same as the distance from b to a!Note: The distance from a to b is the same as the distance from b to a!Distance from a to bDistance from b to aR.2 Algebra Essentials: Find Distance on the Real Number Line; Compute Absolute ValuesExample: Write each of the following as an inequality using absolute value notation: x is more than 3 units from 2Example: Normal human body temperature is F. A temperature x that differs from normal by less than F is considered healthy. Write an inequality using absolute value notation that describes a healthy human body temperature.Determine if F is a healthy or unhealthy body temperature.98.6o1.5o100oR.2 Algebra Essentials: Evaluate Algebraic Expressions and Equations for Given ValuesAlways substitute values into parentheses & follow the order of operations!Always substitute values into parentheses & follow the order of operations!Examples: Evaluate the following for x = -3 and y = -5. 2 31xy--23 4y y-Example: If a rock falls from a height of 20 meters on Earth, the height h (in meters) after x seconds is:What is the height of the rock after falling 1.3 seconds? 220 4.9h x= -When responding to a word-problem be sure, in your answer, to:• mirror the original question using a complete sentence and• include unitsR.2 Algebra Essentials: Evaluate Algebraic Expressions and Equations for Given ValuesR.2 Algebra Essentials: Determine the Domain of a VariableThe set of values that a variable may assume is called the domain of the variable.Let’s begin by addressing some problem areas (in the real number system):R.2 Algebra Essentials: Determine the Domain of a VariableExamples: Find the domains for each of the following using set-builder notation and interval notation.set-builder notationinterval notationset-builder notation interval notation52x -3 5x+ +R.2 Algebra Essentials: Determine the Domain of a VariableExamples: Find the domains for each of the following using set-builder notation and interval notation.set-builder notationinterval notationset-builder notation interval notation219x +29xx -R.2 Algebra Essentials: Use the Laws of Exponents to Evaluate ExpressionsExponents denote repeated multiplicationExample: FindLaws of Exponents: For any integers m, n (assuming no divisions by 0)35m nx x =mnxx=( )nmx =Laws of Exponents examples( )nxy =R.2 Algebra Essentials: Use the Laws of Exponents to Evaluate ExpressionsLaws of Exponents: For any integers m, n (assuming no divisions by 0)Laws of Exponents examples0x =nxy� �=� �� �nx-=1also nx-=R.2 Algebra Essentials: Use the Laws of Exponents to Evaluate ExpressionsExamples: Simplify each of the following; express all answers so that exponents are positive. Whenever an exponent is 0 or negative, we assume the base is not zero.3 9x x( )27x3 5x x-( )35x-R.2 Algebra Essentials: Use the Laws of Exponents to Evaluate ExpressionsExamples: Simplify each of the following; express all answers so that exponents are positive. Whenever an exponent is 0 or negative, we assume the base is not zero.( )342x02x34x� �� �� �25xx--R.2 Algebra Essentials: Use the Laws of Exponents to Evaluate ExpressionsExamples: Simplify each of the following; express all answers so that exponents are positive. Whenever an exponent is 0 or negative, we assume the base is not zero.2 13 44 ( )2x yzx y- -22x yxy-32256xy---� �� �� �R.2 Algebra Essentials: Use the Laws of Exponents to Evaluate ExpressionsExamples: Evaluate using a calculator; round your answer to three decimal places.68.2-( )68.2-( )52.3-R.2 Algebra Essentials: Evaluate Square RootsIf a is a non-negative real number, any number b, such that is the square root of a and is denotedIf a is a non-negative real number, any non-negative number b, such that is the primary square root of a and is denotedExamples: Evaluate the following by taking the square root.2b a=b a=6409164-2b a=b a=The principal root of a positive number is positiveNegative numbers do not have real # square rootsprincipal root:principal root:R.2 Algebra Essentials: Evaluate Square RootsExamples: Find the principal square root of each of the following( )21.42( 3)-2x( )2If 0, then c c c� =2For any real number , c c c=Just in time…R.2 Algebra EssentialsIndependent Practice You learn math by doing math. The best way to learn math is to practice, practice, practice. The assigned homework examples provide you with an opportunity to practice. Be sure to complete every assigned problem (or more if you need additional practice). Check your answers to the odd-numbered problems in the back of the text to see whether you have correctly solved each problem; rework all problems that are incorrect.Read pp. 17-26Homework: pp. 26-29 #7, 15-19 odds, 23, 29, 31, 41, 43, 47-53 odds, 63, 65, 67, 71, 73-107 odds, 111, 113,