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**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.

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**2.4 – Linear Inequalities in One Variable**

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**2.4 – Linear Inequalities in One Variable**

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**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.

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**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.

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**2.4 – Linear Inequalities in One Variable**

Example Solve: Graph the solution set. [

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**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.

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**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. (

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**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 [

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**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 – > – 4x > –18 x > –4.5 (

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**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.

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**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}.

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**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, )

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**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]

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**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]

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**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.

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**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.

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**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, )

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**Compound Inequalities**

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

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**2.4 – Linear Inequalities in One Variable**

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