2.4 – Linear Inequalities in One Variable

1. An inequality is a statement that contains one of the symbols: < , >, ≤ or ≥. Equations Inequalities 2.4 – Linear Inequalities in One Variable 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. 2.4 – Linear Inequalities in One Variable

3. 2.4 – Linear Inequalities in One Variable

4. 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.

5. 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.

6. 2.4 – Linear Inequalities in One Variable Example Solve: Graph the solution set. [

7. 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.

8. 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.

9. 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

10. 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

11. 2.4 – Linear Inequalities in One Variable Intersection of Sets Compound Inequalities The solution set of a compound inequality formed with and is the intersection of the individual solution sets.

12. 2.4 – Linear Inequalities in One Variable Example Find the intersection of: Compound Inequalities The numbers 4 and 6 are in both sets. The intersection is {4, 6}.

13. 2.4 – Linear Inequalities in One Variable Example Solve and graph the solution for x + 4 > 0 and 4x > 0. Compound Inequalities First, solve each inequality separately. 4x > 0 x + 4 > 0 and ( x > – 4 x > 0 (0, )

14. 2.4 – Linear Inequalities in One Variable Example 0  4(5 – x) < 8 Compound Inequalities 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 5 4 5  x > 3 (3,5]

15. 2.4 – Linear Inequalities in One Variable Example – Alternate Method 0  4(5 – x) < 8 4(5 – x) < 8 0  4(5 – x) Compound Inequalities 20 – 4x < 8 0  20 – 4x 20 – 20 – 4x < 8 – 20 0 – 20  20 – 20 – 4x – 4x < – 12 – 20  – 4x Dividing by negative: change sign Dividing by negative: change sign 3 5 4 x > 3 5  x ( [ (3,5]

16. 2.4 – Linear Inequalities in One Variable Union of Sets The solution set of a compound inequality formed with or is the union of the individual solution sets. Compound Inequalities

17. 2.4 – Linear Inequalities in One Variable Example Find the union of: Compound Inequalities The numbers that are in either set are {2, 3, 4, 5, 6, 8}. This set is the union.

18. 2.4 – Linear Inequalities in One Variable Example: Solve and graph the solution for 5(x – 1)  –5 or 5 – x < 11 Compound Inequalities 5(x – 1)  –5 or 5 – x < 11 5x – 5  –5 –x < 6 5x 0 x > – 6 x 0 (–6, )

19. 2.4 – Linear Inequalities in One Variable Example: Compound Inequalities or

20. 2.4 – Linear Inequalities in One Variable