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Chapter 1 Section 4 Solving Inequalities

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ALGEBRA 2 LESSON 1-4 State whether each inequality is true or false. 1.5 < < – – (For help, go to Lessons 1-1 and 1-3.) Solving Inequalities Solve each equation. 7.3x + 3 = 2x – 38.5x = 9(x – 8) + 12 > – < – < – > –

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2.5 < –12, false 4.5 –12, false 6.5 5, true 8.5x = 9(x – 8) x = 9x – –4x = –60 x = 15 Solutions Solving Inequalities 1.5 < 12, true , false 5.5 5, true 7.3x + 3 = 2x – 3 3x – 2x = –3 – 3 x = –6 ALGEBRA 2 LESSON 1-4 > – < – < – > –

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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. ( and > becomes <) When graphing on a number line: Open dot for 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 ALGEBRA 2 LESSON 1-4 Solve –2x < 3(x – 5). Graph the solution. –2x < 3(x – 5) –2x < 3x – 15Distributive Property –5x < –15Subtract 3x from both sides. x > 3Divide each side by –5 and reverse the inequality.

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Try These Problems Solve each inequality. Graph the solution. a) 3x – 6 < 27 a) 3x < 33 x < 11 b) 12 2(3n + 1) + 22 a) 12 6n n n -2 n

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

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Try These Problems Solve. Graph the solution. a) 2x < 2(x + 1) + 3 a) 2x < 2x x < 2x < 5 All Real Numbers b) 4(x – 3) + 7 4x + 1 a) 4x – x + 1 4x x No Solution 0 0

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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? Define:Let x = sales (in dollars). The sales must be greater than or equal to $75,000. Relate:$ % of sales $5000 > – Write: x 5000 > – 0.04x 3000Subtract 2000 from each side. > – x 75,000Divide each side by > –

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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 x 1100 x 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 -1 < 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 a) Graph the solution of 3x – 1 > -28 and 2x + 7 < 19. 3x > -27 and 2x < 12 x > -9 and x < 6 b) Graph the solution of -8 < 3x + 1 <19 -9 < 3x < < 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 < –12 –2x < 4 x –2

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Try These Problems a) Graph the solution of 2x > x + 6 and x – 7 < 2 a) x > 6 and x < 9 b) Graph the solution of x – 1 8 a) x 11

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Homework Practice 1.4 All omit 23

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