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## Solve radical equations and inequalities.

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**Objective**Solve radical equations and inequalities.**Vocabulary**radical equation radical inequality**A radical equationcontains 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.**Remember!**For a square root, the index of the radical is 2.**Example 1A: Solving Equations Containing One Radical**Solve each equation.**Example 1B: Solving Equations Containing One Radical**Solve each equation.**Check It Out! Example 1a**Solve the equation.**Check It Out! Example 1b**Solve the equation.**Check It Out! Example 1c**Solve the equation.**Solve**Example 2: Solving Equations Containing Two Radicals**Check It Out! Example 2a**Solve each equation.**Check It Out! Example 2b**Solve each equation.**Helpful Hint**You can use the intersect feature on a graphing calculator to find the point where the two curves intersect. Raising each side of an equation to an even power may introduce extraneous solutions.**Let Y1 = and Y2 = 5 – x.**Example 3: Solving Equations with Extraneous Solutions Solve . Method 1 Use a graphing calculator. The graphs intersect in only one point, so there is exactly one solution. The solution is x = – 1**Example 3 Continued**Method 2 Use algebra to solve the equation. Step 1 Solve for x. Square both sides. Simplify. –3x + 33 = 25 – 10x + x2 0 = x2 – 7x – 8 Write in standard form. 0 = (x – 8)(x + 1) Factor. x – 8 = 0 or x + 1 = 0 Solve for x. x = 8 or x = –1**Example 3 Continued**Method 2 Use algebra to solve the equation. Step 2 Use substitution to check for extraneous solutions. 3 –3 6 6 x Because x = 8 is extraneous, the only solution is x = –1.**Check It Out! Example 3a**Solve each equation. Method 1 Use a graphing calculator.**Let Y1 = and Y2 = x +3.**Check It Out! Example 3a Solve each equation. Method 1 Use a graphing calculator. The graphs intersect in only one point, so there is exactly one solution. The solution is x = 1.**Check It Out! Example 3a Continued**Method 2 Use algebra to solve the equation.**Check It Out! Example 3a Continued**Method 1 Use algebra to solve the equation. Step 2 Use substitution to check for extraneous solutions. x 2 –2 4 4 Because x = –5 is extraneous, the only solution is x = 1.**Check It Out! Example 3b**Solve each equation.**Let Y1 = and Y2 = –x +4.**Check It Out! Example 3b Solve each equation. Method 1 Use a graphing calculator. The graphs intersect in two points, so there are two solutions. The solutions are x = –4 and x = 3.**Check It Out! Example 3b Continued**Method 2 Use algebra to solve the equation.**Check It Out! Example 3b Continued**Method 1 Use algebra to solve the equation. Step 2 Use substitution to check for extraneous solutions. **Remember!**To find a power, multiply the exponents.**1**3 Example 4A: Solving Equations with Rational Exponents Solve each equation. (5x + 7) = 3**2x = (4x + 8)**1 2 Example 4B: Solving Equations with Rational Exponents**1**1 1 1 1 1 2 2 2 2 2 2 2x = (4x + 8) 2x = (4x + 8) 2(–1) (4(–1) + 8) 4 4 –2 2 x –2 4 4 16 2(2) (4(2) + 8) Example 4B Continued Step 2 Use substitution to check for extraneous solutions. The only solution is x = 2.**1**3 Check It Out! Example 4a Solve each equation. (x + 5) = 3**1**2 Check It Out! Example 4b (2x + 15) = x**1**1 1 1 1 1 2 2 2 2 2 2 (2x + 15) = x (2x + 15) = x (2(5) + 15) 5 (2(–3) + 15) –3 3 –3 x 5 5 (10 + 15) 5 (–6 + 15) –3 Check It Out! Example 4b Continued Step 2 Use substitution to check for extraneous solutions. The only solution is x = 5.**1**2 Check It Out! Example 4c 3(x + 6) = 9**Remember!**A radical expression with an even index and a negative radicand has no real roots. A radical inequality is an inequality that contains a variable within a radical. You can solve radical inequalities by graphing or using algebra.**Solve .**Example 5: Solving Radical Inequalities**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. Example 5: Solving Radical Inequalities Method 1 Use a graph and a table. The solution is 3 ≤ x ≤ 21.**Example 5 Continued**Method 2 Use algebra to solve the inequality.**Solve .**Check It Out! Example 5a**Solve .**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. Check It Out! Example 5a Method 1 Use a graph and a table. The solution is 3 ≤ x ≤ 12.**Check It Out! Example 5a Continued**Method 2 Use algebra to solve the inequality.**Check It Out! Example 5b**Solve .**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. Check It Out! Example 5b Solve . Method 1 Use a graph and a table. The solution is x ≥ –1.**Check It Out! Example 5b Continued**Method 2 Use algebra to solve the inequality.**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.**Lesson Quiz: Part I**Solve each equation or inequality. 1. 2. 3. 4. 5.