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**Section 4 Solving Inequalities**

Chapter 1 Section 4 Solving Inequalities

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**Solving Inequalities State whether each inequality is true or false.**

ALGEBRA 2 LESSON 1-4 (For help, go to Lessons 1-1 and 1-3.) State whether each inequality is true or false. 1. 5 < 12 2. 5 < –12 –12 > – < – < – > – Solve each equation. 7. 3x + 3 = 2x – 3 8. 5x = 9(x – 8) + 12

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**Solving Inequalities Solutions 1. 5 < 12, true 3. 5 12, false**

ALGEBRA 2 LESSON 1-4 Solutions 1. 5 < 12, true , false , true 7. 3x + 3 = 2x – 3 3x – 2x = –3 – 3 x = –6 2. 5 < –12, false –12, false , true 8. 5x = 9(x – 8) + 12 5x = 9x – –4x = –60 x = 15 < – > – > – < –

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**Inequalities The solutions include more than one number**

Ex: 2 < x ;values that x could be include 3, 7, 45… All of the rules for solving equations apply to inequalities, with one added: If you multiply or divide by a NEGATIVE you must FLIP the sign. (< becomes > and > becomes <) When graphing on a number line: Open dot for < or > Closed (solid) dot for ≤ or ≥ The shading should be easy to see (a slightly elevated line is ok) --- see examples

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**Solving Inequalities –2x < 3(x – 5)**

ALGEBRA 2 LESSON 1-4 Solve –2x < 3(x – 5). Graph the solution. –2x < 3(x – 5) –2x < 3x – 15 Distributive Property –5x < –15 Subtract 3x from both sides. x > 3 Divide each side by –5 and reverse the inequality.

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**Try These Problems Solve each inequality. Graph the solution.**

3x – 6 < 27 3x < x < 11 12 ≥ 2(3n + 1) + 22 12 ≥ 6n ≥ 6n ≥ 6n -2 ≥ n

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**Solve 7x ≥ 7(2 + x). Graph the solution.**

7x ≥ x Distributive Property 0 ≥ 14 Subtract 7x from both sides. The last inequality is always false, so 7x ≥ 7(2 + x) is always false. It has no solution.

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**Try These Problems Solve. Graph the solution. 2x < 2(x + 1) + 3**

2x < 2x x < 2x < 5 All Real Numbers 4(x – 3) + 7 ≥ 4x + 1 4x – ≥ 4x x + 12 ≥ 4x ≥ 1 No Solution

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Solving Inequalities A real estate agent earns a salary of $2000 per month plus 4% of the sales. What must the sales be if the salesperson is to have a monthly income of at least $5000? Relate: $ % of sales $5000 > – Define: Let x = sales (in dollars). Write: x > – 0.04x Subtract 2000 from each side. > – x 75,000 Divide each side by 0.04. > – The sales must be greater than or equal to $75,000.

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Try This Problem A salesperson earns a salary of $700 per month plus 2% of the sales. What must the sales be if the salesperson is to have a monthly income of at least $1800? x ≥ 1800 .02x ≥ 1100 x ≥ 55000 The sales must be at least $55,000.

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**Compound Inequalities**

Compound Inequality – a pair of inequalities joined by and or or Ex: -1 < x and x ≤ 3 which can be written as < x ≤ 3 x < -1 or x ≥ 3 For and statements the value must satisfy both inequalities For or statements the value must satisfy one of the inequalities

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And Inequalities Graph the solution of 3x – 1 > -28 and 2x + 7 < 19. 3x > -27 and 2x < 12 x > -9 and x < 6 Graph the solution of -8 < 3x + 1 <19 -9 < 3x < 18 -3 < x < 6

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Or Inequalities ALGEBRA 2 LESSON 1-4 Graph the solution of 3x + 9 < –3 or –2x + 1 < 5. 3x + 9 < –3 or –2x + 1 < 5 3x < – –2x < 4 x < –4 or x > –2

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Try These Problems Graph the solution of 2x > x + 6 and x – 7 < 2 x > 6 and x < 9 Graph the solution of x – 1 < 3 or x + 3 > 8 x < 4 or x > 11

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Homework Practice 1.4 All omit 23

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Main Idea/Vocabulary Solve inequalities by using the Multiplication or Division Properties of Inequality.

Main Idea/Vocabulary Solve inequalities by using the Multiplication or Division Properties of Inequality.

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