170 Chapter 2 Linear Equations and Inequalities in One Variable Multiplying both sides of an inequality by a positive number maintains the inequality relationship. Let c = -1. a < b ac > bc a c > b c -6 < 12 -6(-1) > 12(-1) -6 -1 > 12 -1 True 6 > -12 6 > -12 True True Multiplying both sides of an inequality by a negative number requires us to reverse the inequality symbol to maintain the inequality relationship. ���������������������� Solving a Linear Inequality Step 1: Clear any parentheses from the equation by applying the distributive property. Step 2: Remove any fractions by multiplying by the LCD. Step 3: Use the addition property of inequality to collect all variable terms on one side and all constant terms on the other side. Step 4: Use the multiplication property of inequality to get a coefficient of 1 on the variable. Remember that multiplying or dividing by a negative number, reverses the inequality symbol. Step 5: If the inequality is a compound inequality of the form a < x < b, then the variable must be eliminated in the middle. Any operation that is required to isolate the variable must be done to all three parts of the inequality. Step 6: Graph the solution set. Step 7: Write the solution set in interval notation or set-builder notation. �� ������������������������ �������������������� Solve each inequality. Graph the solution set and write the solution set in interval and set-builder notation. 4a. x + 4 < -1 4b. -4x ≤ 4 4c. 3y - 7 > 2 4d. 3(2a + 1) - a ≥ 2a - 7 4e. 1 2 b - ab + 3 4 b > 1 4 b + 1 2 4f. 2(4x - 3) > 2x + 3(2x + 1) 4g. x + 5(x - 2) < 3(2x + 1) +7 4h. -3 < 2x + 1 < 7 Solutions 4a. x + 4 < -1 x + 4 - 4 < -1 - 4 Subtract 4 from each side. x < -5 Simplify. The graph of the solution set consists of all numbers less than -5; that is, the numbers to the left of -5 but not including -5. –10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 Interval notation: (-∞, -5) Set-builder notation: Exux<-5F
hendricks_beginning_algebra_1e_ch1_3
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