2.4 – Linear Inequalities in One Variable An inequality is a statement that contains one of the symbols: < , >, ≤ or ≥. Equations Inequalities x = 3 x > 3 12 = 7 – 3y 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. b.   c.

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: Graph the solution set.           [      

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: Graph the solution set.       The inequality symbol is reversed since we divided by a negative number. (      

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 (        

Compound Inequalities 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.

Compound Inequalities 2.4 – Linear Inequalities in One Variable Compound Inequalities Example Find the intersection of: The numbers 4 and 6 are in both sets. The intersection is {4, 6}.

Compound Inequalities 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 4x > 0 and x > – 4 x > 0 ( (0, )      

Compound Inequalities 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 Remember that the sign direction changes when you divide by a number < 0! ( 3 4 5 [ 5  x > 3 (3,5]

Compound Inequalities 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 20 – 4x < 8 0 – 20  20 – 20 – 4x 20 – 20 – 4x < 8 – 20 – 20  – 4x – 4x < – 12 Dividing by negative: change sign Dividing by negative: change sign x > 3 5  x ( 3 4 5 [ (3,5]

Compound Inequalities 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.

Compound Inequalities 2.4 – Linear Inequalities in One Variable Compound Inequalities Example Find the union of: The numbers that are in either set are {2, 3, 4, 5, 6, 8}. This set is the union.

Compound Inequalities 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 or 5 – x < 11 5x – 5  –5 –x < 6 5x  0 x > – 6 x  0 (–6, )

Compound Inequalities 2.4 – Linear Inequalities in One Variable Compound Inequalities Example:     or        

2.4 – Linear Inequalities in One Variable