Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Slide 241.5 & 1.6: Absolute Value Equations; InequalitiesChapter 1 Sections 5-6: Absolute Value Equations; InequalitiesIn this section, we will… Solve linear inequalities and compound inequalities Solve equations involving absolute value Solve inequalities involving absolute value Solve applied problems1.5 & 1.6: Solve Linear Inequalities and Compound InequalitiesAn unbounded interval is an interval that extends forever in one direction.inequality notation graph of solution set interval notationall real numbers xx a>x a�x a<x a�aaaaaA linear inequality in one variable (say, x) is any inequality that can be expressed in one of the following forms, where a, b, c are real numbers0a � ax b c ax b c ax b c ax b c+ < + � + > + �1.5 & 1.6: Solve Linear Inequalities and Compound InequalitiesProperty 1 of Inequalities: Any real number can be added (or subtracted) from both sides of an inequality. This produces an equivalent inequality.Property 2 of Inequalities: Both sides of an inequality can be multiplied (or divided) by a positive number. This produces an equivalent inequality.Property 3 of Inequalities: If both sides of an inequality are multiplied (or divided) by a negative number, another inequality results (opposite direction).example: - 4 < 6example: - 4 < 6example: - 4 < 61.5 & 1.6: Solve Linear Inequalities and Compound InequalitiesExample: Solve the inequality. Graph the solution set and express your answer using interval notation.graph1 2 3x- �interval notation1.5 & 1.6: Solve Linear Inequalities and Compound InequalitiesExample: Solve the inequality. Graph the solution set and express your answer using interval notation.graph8 4(2 ) 2x x- - <-interval notation1.5 & 1.6: Solve Linear Inequalities and Compound InequalitiesA compound inequality is a connected pair of inequalities. Note that a double inequality is assumed to be a compound inequality containing the word “and”.inequality notation graph of solution set interval notationopen intervals:half-open intervals:closed intervals:a x b< <a x b� <a x b< �a x b� �aaaai.e. means and c x d c x x d< < < <bbbb1.5 & 1.6: Solve Linear Inequalities and Compound InequalitiesExample: Solve the inequality. Graph the solution set and express your answer using interval notation.graph3 3 2 9x- � - <interval notation1.5 & 1.6: Solve Linear Inequalities and Compound InequalitiesExample: Solve the inequality. Graph the solution set and express your answer using interval notation.graph3 20 42x +< <interval notationHow to Solve a Word Problem:Step 1: Read the problem until you understand it. • What are we asked to find? • What information is given?• What vocabulary is being used?Step 2: Assign a variable to represent what you are looking for. Express any remaining unknown quantities in terms of this variable.Step 3: Make a list of all known facts and form an equation or inequality to solve. It may help to make a labeled: diagram, table or chart, graphStep 4: SolveStep 5: State the solution in a complete sentence by mirroring the original question. Be sure to include units when necessary.Step 6: Check your result(s) in the words of the problem• Does your solution make sense?1.5 & 1.6: Solve Applied Problems1.5 & 1.6: Solve Applied ProblemsExample: The village of Quahog charges homeowners $28.84 per quarter year plus $2.28 per 1000 gallons of water usage in excess of 12,000 gallons. In 2008, the Griffin’s quarterly bill ranged from a high of $74.44 to a low of $42.52. Over what range did the Griffin’s water usage vary?1.5 & 1.6: Solve Equations Involving Absolute ValueSolving Equations Involving Absolute Value:Recall from section 2 of the review chapter, that absolute values measures the distance a given number is from zero. examples:• isolate the absolute value• apply the appropriate rule (boxes above)• solve the two equations and check your resultsgiven that x 7, we know=given that x 7, we know=-If 0, then means or k x k x k x k� = = =-note that 3 3 also 3 3= =- means a or aa b b b= = =-To solve an absolute value equation, you must solve two separate equations!1.5 & 1.6: Solve Equations Involving Absolute ValueExample: Solve the equation and check your results.checks3 1 2x - =1.5 & 1.6: Solve Equations Involving Absolute ValueExample: Solve the equation and check your results.checks1 2 6 9a- + =1.5 & 1.6: Solve Equations Involving Absolute ValueExample: Solve the equation and check your results.checks2 3 4 9x x- = +1.5 & 1.6: Solve Equations Involving Absolute ValueExample: Solve the equation and check your results.checks2 1v- =-1.5 & 1.6: Solve Inequalities Involving Absolute ValueSolving Inequalities Involving Absolute Value:Recall that absolute values measures the distance a given number is from zero. example: in English: graph:example: in English: graph:3x =3x �What if it was strictly less than?What if this was “-3”?What if this was “-3”?1.5 & 1.6: Solve Inequalities Involving Absolute ValueRecall that absolute values measures the distance a given number is from zero. example: in English: graph:Solving Inequalities Involving Absolute Value:•isolate the absolute value• apply the appropriate rule •solve the two inequalities 3x �What if this was “-3”?If 0, then means means means or means or kx k k x kx k k x kx k x k x kx k x k x k�< - < <� - � �> <- >� � �Example: Solve the inequality. Graph the solution set and express your answer using interval notation.graph2 3 1x- >interval notation1.5 & 1.6: Solve Inequalities Involving Absolute ValueExample: Solve the inequality. Graph the solution set and express your answer using interval notation.graph6 12x <-interval notation1.5 & 1.6: Solve Inequalities Involving Absolute ValueExample: Solve the inequality. Graph the solution set and express your answer using interval notation.graph1 2 4 1x- - <-interval notation1.5 & 1.6: Solve Inequalities Involving Absolute ValueExample: Solve the inequality. Graph the solution set and express your answer using interval notation.graph1 2 3x- �-interval notation1.5 & 1.6: Solve Inequalities Involving Absolute ValueExample: Solve the inequality. Graph the solution set and express your