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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

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

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

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

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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.

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

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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

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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

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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

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

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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.

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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

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

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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

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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.

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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

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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.

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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

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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.

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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.

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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

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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

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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.

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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

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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

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

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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

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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.

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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.

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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

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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

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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 ( )

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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.

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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!

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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.

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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

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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.

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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.

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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

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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.

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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.

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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

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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.

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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.

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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.

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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

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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.

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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.

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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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