Algebra I – Inequalities Graphing and Absolute Value - Quick Guides | Mathematics

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Study GuideAlgebra I–InequaliƟes Graphing and Absolute Value1. InequaliƟesWhat Is an Inequality?Aninequalityis a mathematical statement that compares two values that arenot necessarily equal.Instead of using an equals sign (=), inequalities use special symbols to show the relationship betweennumbers.Common inequality symbols include:•>: greater than•<: less than•≥: greater than or equal to•≤: less than or equal toThese symbols help us describe whether one number is larger, smaller, or possibly equal to anothernumber.Axioms and ProperƟes of InequaliƟesWhen working with inequalities, mathematicians follow certainbasic rules (called axioms orproperties).These rules apply toall real numbers(a), (b), and (c).Understanding these properties will help you compare numbers and solve inequality problemscorrectly.1. Trichotomy AxiomTheTrichotomy Axiomstates that for any two numbers,only one of the following can be true:

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Study Guide•(a > b)•(a = b)•(a < b)In simple terms, when comparing two numbers,one number must be greater, equal, or smallerthan the other. No other possibilities exist.2. TransiƟve Property of InequaliƟesThetransitive propertyhelps us compare numbers through a chain of relationships.If:[a > b and b > c]then:[a > c]ExampleIf:[3 > 2 and 2 > 1]then:[3 > 1]The same idea works forless thanrelationships.Example:[4 < 5 and 5 < 6]Therefore:[4 < 6]

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Study Guide3. AddiƟon and SubtracƟon PropertyYou canadd or subtract the same number on both sides of an inequality, and the inequality willremain true.If:[a > b]then:[a + c > b + c]and[a-c > b-c]Similarly:If:[a < b]then:[a + c < b + c]and[a-c < b-c]Key IdeaAdding or subtracting thesame value from both sidesdoesnot change the directionof theinequality.ExamplesIf:[3 > 2]Add 1 to both sides:

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Study Guide[3 + 1 > 2 + 1][4 > 3]Another example:[12 < 15]Subtract 4 from both sides:[12-4 < 15-4][8 < 11]4. MulƟplicaƟon and Division by a PosiƟve NumberWhen youmultiply or divide both sides of an inequality by a positive number, thedirection ofthe inequality stays the same.If:[a > b and c > 0]then:[ac > bcIf:[a < b and c > 0]then:[ac < bc]Important RuleMultiplying or dividing by apositive number does NOT change the inequality sign.ExampleIf:

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Study Guide[5 > 2]Multiply both sides by3:[5(3) > 2(3)][15 > 6]Another example:[3 < 12]Divide both sides by4:5. MulƟplicaƟon and Division by a NegaƟve NumberMultiplying or dividing by anegative number works differently.When you do this, theinequality sign must reverse direction.If:[a > band c < 0]then:[ac < bc]If:[a < b andc < 0]then:[ac > bc]

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Study GuideImportant RuleMultiplying or dividing by anegative number flips the inequality sign.ExampleIf:[5 > 2]Multiply both sides by−3:[5(-3) < 2(-3)][-15 <-6]Notice how the signchanged from > to <.Solving InequalitiesSolving inequalities isvery similar to solving equations.You can:•Add numbers•Subtract numbers•Multiply•Divideon both sides to isolate the variable.However, there isone very important difference:If you multiply or divide by a negative number, you must reverse the inequality sign.Keeping this rule in mind will help you solve inequalities correctly.

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Study GuideSolving Inequalities: Worked ExamplesNow that we understand the rules of inequalities, let’s see how tosolve them step by step.Solving inequalities is very similar to solving equations. We use basic operations such as:•Adding or subtracting numbers•Multiplying or dividing both sidesHowever, remember one important rule:If you multiply or divide by a negative number, you must reverse the inequality sign.Let’s look at some examples to understand this better.Example 1Solve for (x):[2x + 4 > 6]Step 1:Subtract 4 from both sidesWe remove the constant term from the left side.Step 2:Divide both sides by 2Final Answer

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Study GuideThis meansx can be any number greater than 1.Writing the Answer in Set Builder NotationSometimes solutions are written usingset builder notation.For this example:This is read as:“The set of all x such that x is greater than 1.”Example 2Solve for (x):Step 1:Divide both sides by−7Since we are dividing by anegative number, we mustreverse the inequality sign.Step 2:SimplifyFinal Answer

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Study GuideThis meansx can be any number less than−2.Example 3Solve for (x):Step 1:Subtract 2 from both sidesStep 2:Subtract (5x) from both sidesStep 3:Divide both sides by−2Because we divide by anegative number, we mustreverse the inequality sign.Final AnswerThis meansx can be any number less than or equal to 6.

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Study GuideSet Builder NotationThe solution can also be written as:2. Quiz: InequaliƟes1. QuestionWhich of the following symbols represents “is less than or equal to”?Answer Choices•≥•<•≤Correct Answer:≤Why This Is CorrectThe symbol≤means “less than or equal to.”It shows that a value can besmaller than another value or exactly equal to it.Example:Ifx≤5, then x can be5, 4, 3, 2, or any number less than 5.2. QuestionWhich of the following is a false statement?

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