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Holt Algebra 2 8-8 Solving Radical Equations and Inequalities 8-8 Solving Radical Equations and Inequalities Holt Algebra 2 Warm Up Warm Up Lesson Presentation.

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Presentation on theme: "Holt Algebra 2 8-8 Solving Radical Equations and Inequalities 8-8 Solving Radical Equations and Inequalities Holt Algebra 2 Warm Up Warm Up Lesson Presentation."— Presentation transcript:

1 Holt Algebra Solving Radical Equations and Inequalities 8-8 Solving Radical Equations and Inequalities Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson Quiz Lesson Quiz

2 Holt Algebra Solving Radical Equations and Inequalities Warm Up Simplify each expression. Assume all variables are positive. Write each expression in radical form

3 Holt Algebra Solving Radical Equations and Inequalities Solve radical equations and inequalities. Objective

4 Holt Algebra Solving Radical Equations and Inequalities radical equation radical inequality Vocabulary

5 Holt Algebra Solving Radical Equations and Inequalities A radical equation contains a variable within a radical. Recall that you can solve quadratic equations by taking the square root of both sides. Similarly, radical equations can be solved by raising both sides to a power.

6 Holt Algebra Solving Radical Equations and Inequalities For a square root, the index of the radical is 2. Remember!

7 Holt Algebra Solving Radical Equations and Inequalities Solve each equation. Example 1A: Solving Equations Containing One Radical Subtract 5. Simplify. Square both sides. Solve for x. Simplify. Check

8 Holt Algebra Solving Radical Equations and Inequalities 7 Example 1B: Solving Equations Containing One Radical Divide by 7. Simplify. Cube both sides. Solve for x. Simplify. Check Solve each equation x  

9 Holt Algebra Solving Radical Equations and Inequalities Solve the equation. Subtract 4. Simplify. Square both sides. Solve for x. Simplify. Check Check It Out! Example 1a

10 Holt Algebra Solving Radical Equations and Inequalities Cube both sides. Solve for x. Simplify. Check Check It Out! Example 1b Solve the equation.

11 Holt Algebra Solving Radical Equations and Inequalities Divide by 6. Square both sides. Solve for x. Simplify. Check Check It Out! Example 1c Solve the equation.

12 Holt Algebra Solving Radical Equations and Inequalities Example 2: Solving Equations Containing Two Radicals Square both sides. Solve Simplify. Distribute. Solve for x. 7x + 2 = 9(3x – 2) 7x + 2 = 27x – = 20x 1 = x

13 Holt Algebra Solving Radical Equations and Inequalities Example 2 Continued Check 3

14 Holt Algebra Solving Radical Equations and Inequalities Square both sides. Solve each equation. Simplify. Solve for x. 8x + 6 = 9x 6 = x Check It Out! Example 2a Check

15 Holt Algebra Solving Radical Equations and Inequalities Cube both sides. Solve each equation. Simplify. Distribute. x + 6 = 8(x – 1) Check It Out! Example 2b x + 6 = 8x – 8 14 = 7x 2 = x Check 2 Solve for x.

16 Holt Algebra Solving Radical Equations and Inequalities Raising each side of an equation to an even power may introduce extraneous solutions. You can use the intersect feature on a graphing calculator to find the point where the two curves intersect. Helpful Hint

17 Holt Algebra Solving Radical Equations and Inequalities Solve. Example 3: Solving Equations with Extraneous Solutions Method 1 Use a graphing calculator. The solution is x = – 1 The graphs intersect in only one point, so there is exactly one solution. Let Y1 = and Y2 = 5 – x.

18 Holt Algebra Solving Radical Equations and Inequalities Example 3 Continued Method 2 Use algebra to solve the equation. Step 1 Solve for x. Square both sides. Solve for x. Factor. Write in standard form. Simplify. –3x + 33 = 25 – 10x + x 2 0 = x 2 – 7x – 8 0 = (x – 8)(x + 1) x – 8 = 0 or x + 1 = 0 x = 8 or x = –1

19 Holt Algebra Solving Radical Equations and Inequalities Example 3 Continued Method 2 Use algebra to solve the equation. Step 2 Use substitution to check for extraneous solutions. 3 –36 x Because x = 8 is extraneous, the only solution is x = – 1.

20 Holt Algebra Solving Radical Equations and Inequalities Method 1 Use a graphing calculator. The solution is x = 1. The graphs intersect in only one point, so there is exactly one solution. Check It Out! Example 3a Solve each equation. Let Y1 = and Y2 = x +3.

21 Holt Algebra Solving Radical Equations and Inequalities Method 2 Use algebra to solve the equation. Step 1 Solve for x. Square both sides. Solve for x. Factor. Write in standard form. Simplify. 2x + 14 = x 2 + 6x = x 2 + 4x – 5 0 = (x + 5)(x – 1) x + 5 = 0 or x – 1 = 0 x = –5 or x = 1 Check It Out! Example 3a Continued

22 Holt Algebra Solving Radical Equations and Inequalities Method 1 Use algebra to solve the equation. Step 2 Use substitution to check for extraneous solutions. 4 x Because x = – 5 is extraneous, the only solution is x = 1. Check It Out! Example 3a Continued 2 –2

23 Holt Algebra Solving Radical Equations and Inequalities Method 1 Use a graphing calculator. Check It Out! Example 3b Let Y1 = and Y2 = –x +4. The solutions are x = –4 and x = 3. The graphs intersect in two points, so there are two solutions. Solve each equation.

24 Holt Algebra Solving Radical Equations and Inequalities Method 2 Use algebra to solve the equation. Step 1 Solve for x. Square both sides. Solve for x. Factor. Write in standard form. Simplify. –9x + 28 = x 2 – 8x = x 2 + x – 12 0 = (x + 4)(x – 3) x + 4 = 0 or x – 3 = 0 x = –4 or x = 3 Check It Out! Example 3b Continued

25 Holt Algebra Solving Radical Equations and Inequalities Method 1 Use algebra to solve the equation. Step 2 Use substitution to check for extraneous solutions. Check It Out! Example 3b Continued

26 Holt Algebra Solving Radical Equations and Inequalities To find a power, multiply the exponents. Remember!

27 Holt Algebra Solving Radical Equations and Inequalities Example 4A: Solving Equations with Rational Exponents Solve each equation. (5x + 7) = 3 Cube both sides. Solve for x. Factor. Write in radical form. Simplify. 5x + 7 = 27 5x = 20 x = 4 1 3

28 Holt Algebra Solving Radical Equations and Inequalities Example 4B: Solving Equations with Rational Exponents Factor. Solve for x. Factor out the GCF, 4. Raise both sides to the reciprocal power. Simplify. 2x = (4x + 8) 1 2 (2x) 2 = [(4x + 8) ] x 2 = 4x + 8 Write in standard form. 4x 2 – 4x – 8 = 0 4(x 2 – x – 2) = 0 4(x – 2)(x + 1 ) = 0 4 ≠ 0, x – 2 = 0 or x + 1 = 0 x = 2 or x = – 1 Step 1 Solve for x.

29 Holt Algebra Solving Radical Equations and Inequalities Example 4B Continued Step 2 Use substitution to check for extraneous solutions. 2x = (4x + 8) 1 2 2(2) (4(2) + 8) x = (4x + 8) 1 2 2(–1) (4(–1) + 8) 1 2 – –2 2 x The only solution is x = 2.

30 Holt Algebra Solving Radical Equations and Inequalities Solve each equation. (x + 5) = 3 Cube both sides. Solve for x. Write in radical form. Simplify. x + 5 = 27 x = Check It Out! Example 4a

31 Holt Algebra Solving Radical Equations and Inequalities Factor. Solve for x. Raise both sides to the reciprocal power. Simplify. (2x + 15) = x 1 2 [(2x + 15) ] 2 = (x) x + 15 = x 2 Write in standard form. x 2 – 2x – 15 = 0 (x – 5)(x + 3) = 0 x – 5 = 0 or x + 3 = 0 x = 5 or x = – 3 Step 1 Solve for x. Check It Out! Example 4b

32 Holt Algebra Solving Radical Equations and Inequalities The only solution is x = 5. Check It Out! Example 4b Continued (2(–3) + 15) –3 1 2 (–6 + 15) –3 1 2 x (2x + 15) = x –3 Step 2 Use substitution to check for extraneous solutions. (2(5) + 15) (2x + 15) = x 1 2 ( )

33 Holt Algebra Solving Radical Equations and Inequalities Solve for x. Raise both sides to the reciprocal power. Simplify. 9(x + 6) = 81 [3(x + 6) ] 2 = (9) Distribute 9. 9x + 54 = 81 Check It Out! Example 4c 9x = 27 x = 3 3(x + 6) = Simplify.

34 Holt Algebra Solving Radical Equations and Inequalities A radical inequality is an inequality that contains a variable within a radical. You can solve radical inequalities by graphing or using algebra. A radical expression with an even index and a negative radicand has no real roots. Remember!

35 Holt Algebra Solving Radical Equations and Inequalities Example 5: Solving Radical Inequalities Method 1 Use a graph and a table. Solve. On a graphing calculator, let Y1 = and Y2 = 9. The graph of Y1 is at or below the graph of Y2 for values of x between 3 and 21. Notice that Y1 is undefined when < 3. The solution is 3 ≤ x ≤ 21.

36 Holt Algebra Solving Radical Equations and Inequalities Example 5 Continued Method 2 Use algebra to solve the inequality. Step 1 Solve for x. Subtract 3. Solve for x. Simplify. Square both sides. 2x ≤ 42 x ≤ 21

37 Holt Algebra Solving Radical Equations and Inequalities Example 5 Continued Method 2 Use algebra to solve the inequality. Step 2 Consider the radicand. The radicand cannot be negative. Solve for x. 2x – 6 ≥ 0 2x ≥ 6 x ≥ 3 The solution of is x ≥ 3 and x ≤ 21, or 3 ≤ x ≤ 21.

38 Holt Algebra Solving Radical Equations and Inequalities Method 1 Use a graph and a table. Check It Out! Example 5a On a graphing calculator, let Y1 = and Y2 = 5. The graph of Y1 is at or below the graph of Y2 for values of x between 3 and 12. Notice that Y1 is undefined when < 3. The solution is 3 ≤ x ≤ 12. Solve.

39 Holt Algebra Solving Radical Equations and Inequalities Method 2 Use algebra to solve the inequality. Step 1 Solve for x. Subtract 2. Solve for x. Simplify. Square both sides. x – 3 ≤ 9 x ≤ 12 Check It Out! Example 5a Continued

40 Holt Algebra Solving Radical Equations and Inequalities Method 2 Use algebra to solve the inequality. Step 2 Consider the radicand. The radicand cannot be negative. Solve for x. x – 3 ≥ 0 x ≥ 3 Check It Out! Example 5a Continued The solution of is x ≥ 3 and x ≤ 12, or 3 ≤ x ≤ 12.

41 Holt Algebra Solving Radical Equations and Inequalities Method 1 Use a graph and a table. Check It Out! Example 5b Solve. The solution is x ≥ –1. On a graphing calculator, let Y1 = and Y2 = 1. The graph of Y1 is at or above the graph of Y2 for values of x greater than –1. Notice that Y1 is undefined when < –2.

42 Holt Algebra Solving Radical Equations and Inequalities Method 2 Use algebra to solve the inequality. Step 1 Solve for x. Solve for x. Cube both sides. x + 2 ≥ 1 x ≥ – 1 Check It Out! Example 5b Continued

43 Holt Algebra Solving Radical Equations and Inequalities Method 2 Use algebra to solve the inequality. Step 2 Consider the radicand. The radicand cannot be negative. Solve for x. x + 2 ≥ 1 x ≥ –1 Check It Out! Example 5b Continued The solution of is x ≥ – 1.

44 Holt Algebra Solving Radical Equations and Inequalities Example 6: Automobile Application The time t in seconds that it takes a car to travel a quarter mile when starting from a full stop can be estimated by using the formula, where w is the weight of the car in pounds and P is the power delivered by the engine in horsepower. If the quarter-mile time from a 3590 lb car is 13.4 s, how much power does its engine deliver? Round to the nearest horsepower.

45 Holt Algebra Solving Radical Equations and Inequalities Example 6 Continued Use the formula to determine the amount of horsepower the 3590 lb car has if it finishes the quarter-mile in 13.4s. Substitute 13.4 for t and 3590 for w. Cube both sides. Solve for P. Simplify P ≈ 709, P ≈ 295 The engine delivers a power of about 295 hp.

46 Holt Algebra Solving Radical Equations and Inequalities A car skids to a stop on a street with a speed limit of 30 mi/h. The skid marks measure 35 ft, and the coefficient of friction was 0.7. Was the car speeding? Explain. Check It Out! Example 6 Use the formula to determine the greatest possible length of the driver’s skid marks if he was traveling 30 mi/h. The speed s in miles per hour that a car is traveling when it goes into a skid can be estimated by using the formula s =, where f is the coefficient of friction and d is the length of the skid marks in feet. 30fd

47 Holt Algebra Solving Radical Equations and Inequalities Check It Out! Example 6 Continued 900 = 21d 43 ≈ d If the car were traveling 30 mi/h, its skid marks would have measured about 43 ft. Because the actual skid marks measure less than 43 ft, the car was not speeding. Substitute 30 for s and 0.7 for f. Simplify. Square both sides. Simplify. Solve for d.

48 Holt Algebra Solving Radical Equations and Inequalities Lesson Quiz: Part I Solve each equation or inequality. x = x = –5 x = 14 x = 2 x ≥ 59 5.

49 Holt Algebra Solving Radical Equations and Inequalities Lesson Quiz: Part II 137,258 ft 3 6. The radius r in feet of a spherical water tank can be determined by using the formula, where V is the volume of the tank in cubic feet. To the nearest cubic foot, what is the volume of a spherical tank with a radius of 32 ft?


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