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NAME DATE 7-2 PERIOD Study Guide and Intervention Solving Exponential Equations and Inequalities Solve Exponential Equations All the properties of rational exponents that you know also apply to real exponents. Remember that am · an = am + n, (am)n = amn, and am ÷ an = am - n. Property of Equality for Exponential Functions If b is a positive number other than 1, then bx = by if and only if x = y. Example 2 Write an exponential function whose graph passes through the points (0, 3) and (4, 81). Example 1 Solve 4 x - 1 = 2 x + 5. 4x - 1 = 2x + 5 Original equation 2 x-1 (2 ) = 2x + 5 Rewrite 4 as 22. 2(x - 1) = x + 5 Prop. of Inequality for Exponential Distributive Property The y-intercept is (0, 3), so a = 3. Since the 4 81 −. other point is (4, 81), b = Subtract x and add 2 to each side. Simplifying Functions 2x - 2 = x + 5 x=7 √3 81 − = √ 27 ≈ 2.280, the equation √ 3 4 4 is y = 3(2.280)x. Exercises Solve each equation. 3 4. 4 x + 1 = 8 2x + 3 7 -− 4 7. 9 x + 1 = 27 x + 4 -10 1 -− 2. 23x = 4x + 2 4 3. 3 2x - 1 = −19 1 5. 8x - 2 = − 16 2 − 6. 25 2x = 125 x + 2 6 3 8. 362x + 4 = 216 x+5 ( 64 ) 1 9. − x-2 2 = 16 3x + 1 4 − 7 9 Write an exponential function for the graph that passes through the given points. 10. (0, 4) and (2, 36) y = 4(3)x 13. (0, 2) and (5, 486) x y = 2(3) 16. (0, 3) and (3, 24) y = 3(2)x Chapter 7 11. (0, 6) and (1, 81) y = 6(13.5)x ( y = 5(2)x 27 14. (0, 8) and 3, − () 3 y=8 − 4 12. (0, 5) and (6, 320) 8 ) 15. (0, 1) and (4, 625) x y = (5)x 17. (0, 12) and (4, 144) 18. (0, 9) and (2, 49) y = 12(1.861)x y = 9(2.333)x 12 Glencoe Algebra 2 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 1. 3 2x - 1 = 3 x + 2 NAME DATE 7-2 Study Guide and Intervention PERIOD (continued) Solving Exponential Equations and Inequalities Solve Exponential Inequalities An exponential inequality is an inequality involving exponential functions. Property of Inequality for Exponential Functions 1 52x - 1 > − 125 2x - 1 5 > 5-3 2x - 1 > -3 2x > -2 x > -1 1 Solve 52x - 1 > − . 125 Original inequality 1 Rewrite − as 5-3. 125 Prop. of Inequality for Exponential Functions Add 1 to each side. Divide each side by 2. Lesson 7-2 Example If b > 1 then bx > by if and only if x > y and bx < by if and only if x < y. The solution set is {x | x > -1}. Exercises Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Solve each inequality. 1 1. 3 x - 4 < − 27 x<1 3. 5 2x < 125 x - 5 5 x>− x > 15 3 4. 10 4x + 1 > 100 x - 2 5 x > -− 2 7. 16 ≥ 4x + 5 x ≤ -3 2x - 4 1 10. − <9 81 5. 7 3x < 49 1 - x 6. 8 2x - 5 < 4 x + 8 31 x<− 2 x<− 5 4 ( 27 ) 1 8. − 2x + 1 ( 243 ) 1 ≤ − 3x - 2 13 x≤− ( 25 11. 323x - 4 > 1284x + 3 1 x ≥ -− 13 x-3 ( 343 ) 7 14. − x-3 > 82 x 7 12. 27 2x - 5 1 5x < − ) (9 15 x<− 16 13 3x + 1 1 2x - 1 13. − ≤ 125 ) (2) 1 9. − 3 x<− 9 41 x < -− x>1 Chapter 7 2. 42x - 2 > 2 x + 1 ( 49 2x + 1 1 ≥ − ) x ≥ -4 ( 27 6x - 1 9 15. − ) 27 -x + 6 ≥ − ) (9 x ≤ -1 13 Glencoe Algebra 2