Section 2.4 Linear Inequalities and Applications 89 So, adding the same number to each side of an inequality or subtracting the same number from each side of an inequality maintains the inequality relationship. Property: Multiplication Property of Inequality If a, b, and c are real numbers and c is positive, then a < b and ac < bc and a c < b c are equivalent inequalities. c is negative, then a < b and ac > bc and a c > b c are equivalent inequalities. This property states that • Multiplying or dividing each side of an inequality by a positive number produces an equivalent inequality. • Multiplying or dividing each side of an inequality by a negative number produces an equivalent inequality only if the inequality symbol is reversed. For an illustration of the property, let a=-6 and b = 12. Let c = 3. a < b ac < bc -6 < 12 -6(3) < 12(3) True -18 < 36 True Multiplying each side of an inequality by a positive number maintains the inequality relationship. Let c=-1. a < b ac > bc -6 < 12 -6(-1) > 12(-1) True 6 > -12 True Multiplying each side of an inequality by a negative number requires us to reverse the inequality symbol to maintain the inequality relationship. Procedure: Solving a Linear Inequality Step 1: a. Clear any parentheses from the equation by applying the distributive property. b. Remove any fractions by multiplying each side by the LCD. Step 2: Use the addition property of inequality to collect all variable terms on one side and all constant terms on the other side. Step 3: Use the multiplication property of inequality to get a coefficient of 1 on the variable. Remember that if we multiply or divide by a negative number, we must reverse the inequality symbol. Step 4: Graph the solution set and write the solution set in interval notation and set-builder notation. a c < b c -6 3 < 12 3 -2 < 4 True a c > b c -6 -1 > 12 -1 6 > -12 True
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