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A Quick Algebra Review1.Simplifying Expressions2.Solving Equations3.Problem Solving4.Inequalities5.Absolute Values6.Linear Equations7.Systems of Equations8.Laws of Exponents9.Quadratics10.Rationals11.RadicalsSimplifying ExpressionsAnexpressionis a mathematical “phrase.”Expressions contain numbersand variables, but not an equal sign.Anequationhas an “equal” sign.Forexample:Expression:Equation:5 + 35 + 3 = 8x + 3x + 3 = 8(x + 4)(x–2)(x + 4)(x–2) = 10x² + 5x + 6x² + 5x + 6 = 0x–8x–8 > 3When wesimplifyan expression, we work until there are as few terms aspossible.This process makes the expression easier to use,(that’s why it’scalled“simplify”).The first thing we want to do when simplifying anexpression is tocombine like terms.
For example:Now you try: x² + 5x + 3x² + x³ - 5 + 3[You should get x³ + 4x² + 5x–2]Order of OperationsPEMDAS–Please Excuse My Dear Aunt Sally, remember that fromAlgebra class?It tells the order in which we can complete operations whensolving an equation.First, complete any work inside PARENTHESIS, thenevaluate EXPONENTS if there are any.Next MULTIPLY or DIVIDEnumbers before ADDING or SUBTRACTING.For example:Insidethe parenthesis,look for more order ofoperation rules -PEMDAS.Wedon’t have anyexponents, but we doneed to multiplybefore we subtract,then add inside theparenthesesbeforewemultiply by negative 2on the outside.Simplify:-2[3 - (-2)(6)]= -2[3-(-12)]= -2[3+12]= -2= -30Simplify:x² + 10x–6–5x + 4= x² + 5x–6 + 4= x² + 5x–2There are many terms to lookat! Let’s start with x².Thereare no other terms with x² inthem, so we move on.10xand 5x are like terms, so weadd theircoefficientstogether.10 + (-5) = 5, so wewrite5x.-6 and 4 are alsolike terms, so we can combinethem to get-2.Isn’t thesimplified expression muchnicer?
Let’s try another one…Practice makes perfect…Now you try:2x + 4 [2–(5x–3)][you should get -18x +20]Simplify:(-4)2+ 2[12 + (3-5)]= (-4)2+ 2[12 + (-2)]= (-4)2+ 2= 16 + 2= 16 + 20= 36Insidethe parenthesis, look fororder of operation rules -PEMDAS.We need to subtract 5 from 3 thenadd 12 inside the parentheses.Thistakes care of the P in PEMDAS,now for the E, Exponents.Wesquare -4.Make sure to use (-4)2ifyou are relying on your calculator.If you input -42the calculator willevaluate the expression usingPEMDAS.It will do the exponentfirst, then multiply by -1, givingyou-16, though we know theanswer is16.Now we can multiplyand then add to finish up.Simplify:(5a2–3a +1)–(2a2–4a+ 6)= (5a2–3a +1)–1(2a2–4a+ 6)= (5a2–3a +1)–1(2a2)–(-1)(- 4a)+(-1)( 6)= (5a2–3a +1)–2a2+4a–6= 5a2–3a +1–2a2+4a–6= 3a2+ a- 5Since there are no like termsinside the parenthesis, weneed to distribute the negativesign and then see what wehave.There is really a -1there but we’re basically lazywhen it comes to the numberone and don’talways write it(since 1 times anything isitself).So we need to take -1timesEVERYTHINGin theparenthesis, not just the firstterm.Once we have donethat, we can combine liketerms and rewrite theexpression.