Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
2.4 – Linear Inequalities in One Variable An inequality is a statement that contains one of the symbols: < , >, ≤ or ≥. Equations Inequalities x=3 12 = 7 – 3y x>3 12 ≤ 7 – 3y A solution of an inequality is a value of the variable that makes the inequality a true statement. The solution set of an inequality is the set of all solutions. 2.4 – Linear Inequalities in One Variable 2.4 – Linear Inequalities in One Variable 2.4 – Linear Inequalities in One Variable Example Graph each set on a number line and then write it in interval notation. a. {x | x 2} b. {x | x 1} c. {x | 0.5 x 3} a. [2, ) b. c. (0.5, 3] 2.4 – Linear Inequalities in One Variable Addition Property of Inequality If a, b, and c are real numbers, then a < b and a + c < b + c a > b and a + c > b + c are equivalent inequalities. Also, If a, b, and c are real numbers, then a < b and a - c < b - c a > b and a - c > b - c are equivalent inequalities. 2.4 – Linear Inequalities in One Variable Example Solve: 3x 4 2 x 6 Graph the solution set. {x | x 10} or 10, [ 2.4 – Linear Inequalities in One Variable Multiplication Property of Inequality If a, b, and c are real numbers, and c is positive, then a < b and ac < bc are equivalent inequalities. If a, b, and c are real numbers, and c is negative, then a < b and ac > bc are equivalent inequalities. The direction of the inequality sign must change when multiplying or dividing by a negative value. 2.4 – Linear Inequalities in One Variable Example Solve: 2.3x 6.9 Graph the solution set. The inequality symbol is reversed since we divided by a negative number. {x | x 3} or ( 3, 2.4 – Linear Inequalities in One Variable Solve: 3x + 9 ≥ 5(x – 1). Graph the solution set. 3x + 9 ≥ 5(x – 1) 3x + 9 ≥ 5x – 5 3x – 3x + 9 ≥ 5x – 3x – 5 9 ≥ 2x – 5 9 + 5 ≥ 2x – 5 + 5 14 ≥ 2x 7≥x x≤7 [ 2.4 – Linear Inequalities in One Variable Example Solve: 7(x – 2) + x > –4(5 – x) – 12. Graph the solution set. 7(x – 2) + x > –4(5 – x) – 12 7x – 14 + x > –20 + 4x – 12 8x – 14 > 4x – 32 8x – 4x – 14 > 4x – 4x – 32 4x – 14 > –32 4x – 14 + 14 > –32 + 14 4x > –18 x > –4.5 ( 2.4 – Linear Inequalities in One Variable Compound Inequalities Intersection of Sets The solution set of a compound inequality formed with and is the intersection of the individual solution sets. 2.4 – Linear Inequalities in One Variable Compound Inequalities Example Find the intersection of: {2, 4,6,8} {3, 4,5,6} The numbers 4 and 6 are in both sets. The intersection is {4, 6}. 2.4 – Linear Inequalities in One Variable Compound Inequalities Example Solve and graph the solution for x + 4 > 0 and 4x > 0. First, solve each inequality separately. x+4>0 x>–4 and 4x > 0 x>0 ( ( -4 0 ( (0, ) 2.4 – Linear Inequalities in One Variable Compound Inequalities Example 0 4(5 – x) < 8 0 20 – 4x < 8 0 – 20 20 – 20 – 4x < 8 – 20 – 20 – 4x < – 12 5x>3 [ Remember that the sign direction changes when you divide by a number < 0! ( 3 4 (3,5] 5 2.4 – Linear Inequalities in One Variable Compound Inequalities Example – Alternate Method 0 4(5 – x) < 8 0 4(5 – x) 4(5 – x) < 8 0 20 – 4x 0 – 20 20 – 20 – 4x 20 – 4x < 8 20 – 20 – 4x < 8 – 20 – 20 – 4x – 4x < – 12 Dividing by negative: change sign Dividing by negative: change sign x>3 5x [ ( 3 4 5 (3,5] 2.4 – Linear Inequalities in One Variable Compound Inequalities Union of Sets The solution set of a compound inequality formed with or is the union of the individual solution sets. 2.4 – Linear Inequalities in One Variable Compound Inequalities Example Find the union of: {2, 4,6,8} {3, 4,5,6} The numbers that are in either set are {2, 3, 4, 5, 6, 8}. This set is the union. 2.4 – Linear Inequalities in One Variable Compound Inequalities Example: Solve and graph the solution for 5(x – 1) –5 or 5 – x < 11 5(x – 1) –5 5x – 5 –5 5x 0 x0 [ 0 ( -6 or 5 – x < 11 –x < 6 x>–6 ( -6 (–6, ) 2.4 – Linear Inequalities in One Variable Compound Inequalities Example: or , 2.4 – Linear Inequalities in One Variable