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§ 2.8 Solving Linear Inequalities

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Martin-Gay, Beginning and Intermediate Algebra, 4ed 22 Linear Inequalities in One Variable A linear inequality in one variable is an inequality that can be written in the form ax + b < c where a, b, and c are real numbers and a is not 0. This definition and all other definitions, properties and steps in this section also hold true for the inequality symbols >, , or .

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Martin-Gay, Beginning and Intermediate Algebra, 4ed 33 Graphing Solutions to Linear Inequalities Use a number line. Use a closed circle at the endpoint of an interval if you want to include the point. Use an open circle at the endpoint if you DO NOT want to include the point. Represents the set {x x 7}. Represents the set {x x > – 4}. Solutions to Linear Inequalities Interval notation, is used to write solution sets of inequalities. Use a parenthesis if you want to include the number Use a bracket if you DO NOT want to include the number..

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Martin-Gay, Beginning and Intermediate Algebra, 4ed 44 Solutions to Linear Inequalities x < 3 -5-4-3-2012345 –1.5 x 3 –2 < x < 0 -5-4-3-2012345-5-4-3-2012345 Interval notation: (–∞, 3) Interval notation: (–2, 0) Interval notation: [–1.5, 3) NOT included Included

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Martin-Gay, Beginning and Intermediate Algebra, 4ed 55 Addition Property of Inequality If a, b, and c are real numbers, then a < b and a + c < b + c are equivalent inequalities. Multiplication Property of Inequality 1. If a, b, and c are real numbers, and c is positive, then a < b and ac < bc are equivalent inequalities. 2. If a, b, and c are real numbers, and c is negative, then a bc are equivalent inequalities. Properties of Inequality

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Martin-Gay, Beginning and Intermediate Algebra, 4ed 66 Solving Linear Inequalities in One Variable 1)Clear the inequality of fractions by multiplying both sides by the LCD of all fractions of the inequality. 2)Remove grouping symbols by using the distributive property. 3)Simplify each side of equation by combining like terms. 4)Write the inequality with variable terms on one side and numbers on the other side by using the addition property of equality. 5)Get the variable alone by using the multiplication property of equality. Solving Linear Inequalities

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Martin-Gay, Beginning and Intermediate Algebra, 4ed 77 Solving Linear Inequalities Example: Solve the inequality. Graph the solution and give your answer in interval notation. 2(x – 3) < 4x + 10 2x – 6 < 4x + 10 Distribute. – 6 < 2x + 10 Subtract 2x from both sides. – 16 < 2x Subtract 10 from both sides. – 8 – 8 Divide both sides by 2. -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 (–8, ∞)

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Martin-Gay, Beginning and Intermediate Algebra, 4ed 88 Since 0 is always greater than –7, the solution is all real numbers. (Any value we put in for x in the original statement will give us a true inequality.) Solving Linear Inequalities Example: Solve the inequality. Graph the solution and give your answer in interval notation. x + 5 x – 2 x x – 7 Subtract 5 from both sides. 0 – 7 Subtract x from both sides. -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 (Always true!) (–∞ ∞)

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Martin-Gay, Beginning and Intermediate Algebra, 4ed 99 Solving Linear Inequalities Example: Solve the inequality. Give your answer in interval notation. a.) 9 < z + 5 < 13 b.) –7 < 2p – 3 ≤ 5 a.) 9 < z + 5 < 13 4 < z < 8 Subtract 5 from all three parts. (4, 8) b.) –7 < 2p – 3 ≤ 5 –4 < 2p ≤ 8Add 3 to all three parts. –2 < p ≤ 4Divide all three parts by 2. (–2, 4]

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Martin-Gay, Beginning and Intermediate Algebra, 4ed 10 3x + 9 5(x – 1) 3x + 9 5x – 5 Use distributive property on right side. 3x – 3x + 9 5x – 3x – 5 Subtract 3x from both sides. 9 2x – 5 Simplify both sides. 14 2x Simplify both sides. 7 x Divide both sides by 2. 9 + 5 2x – 5 + 5 Add 5 to both sides. Solution: Solving Linear Inequalities Example: Solve the inequality. Give your answer in graph form.

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Martin-Gay, Beginning and Intermediate Algebra, 4ed 11 A compound inequality contains two inequality symbols. To solve the compound inequality, perform operations simultaneously to all three parts of the inequality (left, middle and right). Compound Inequalities 0 4(5 – x) < 8 This means 0 4(5 – x) and 4(5 – x) < 8.

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Martin-Gay, Beginning and Intermediate Algebra, 4ed 12 The solution is (3,5]. – 20 – 4x < – 12 Simplify each part. 5 x > 3 Divide each part by –4. Remember that the sign changes direction when you divide by a negative number. 0 – 20 20 – 20 – 4x < 8 – 20 Subtract 20 from each part. Solving Compound Inequalities Example: Solve the inequality. Give your answer in interval notation. 0 20 – 4x < 8 Use the distributive property. 0 20 – 4x < 8

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Martin-Gay, Beginning and Intermediate Algebra, 4ed 13 Inequality Applications Example: Six times a number, decreased by 2, is at least 10. Find the number. 1.) UNDERSTAND Let x = the unknown number. “Six times a number” translates to 6x, “decreased by 2” translates to 6x – 2, “is at least 10” translates ≥ 10. Continued

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Martin-Gay, Beginning and Intermediate Algebra, 4ed 14 decreased – Six times a number 6x6x by 2 2 is at least ≥ 10 Finding an Unknown Number Example continued: 2.) TRANSLATE Continued

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Martin-Gay, Beginning and Intermediate Algebra, 4ed 15 Finding an Unknown Number Example continued: 3.) SOLVE 6x – 2 ≥ 10 6x ≥ 12 Add 2 to both sides. x ≥ 2Divide both sides by 6. 4.) INTERPRET Check: Replace “number” in the original statement of the problem with a number that is 2 or greater. Six times 2, decreased by 2, is at least 10 6(2) – 2 ≥ 10 10 ≥ 10State: The number is 2.

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