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Warm up 1. Solve 2. Solve 3. Decompose to partial fractions

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Lesson 4-7 Radical Equations and Inequalities Objective: To solve radical equations and inequalities

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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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For a square root, the index of the radical is 2. Remember!

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Solve Radical Equations Solve. Add 2 to each side. Find the squares. Square each side to eliminate the radical. Add 1 to each side to isolate the radical. Original equation Example 1

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Solve Radical Equations Original equation Answer: The solution checks. The solution is 38. Check Simplify. Replace y with 38. ? Example 1

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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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Method 1 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 + 9 0 = x 2 + 4x – 5 0 = (x + 5)(x – 1) x + 5 = 0 or x – 1 = 0 x = –5 or x = 1 Example 2

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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. 2 –2 Example 2

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Method 2 Use a graphing calculator. The solution is x = 1. The graphs intersect in only one point, so there is exactly one solution. Solve the equation. Let Y1 = and Y2 = x +3. Example 2

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Practice Solve

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Solve: Practice

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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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Method 1 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 Example 3 Solve.

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Method 1 Use algebra to solve the inequality. Step 2 Consider the radicand. The radicand cannot be negative. Solve for x. x – 3 ≥ 0 x ≥ 3 The solution of is x ≥ 3 and x ≤ 12, or 3 ≤ x ≤ 12. Example 3

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Method 2 Use a graph and a table. 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. Example 3

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Method 1 Use algebra to solve the inequality. Step 1 Solve for x. Solve for x. Cube both sides. x + 2 ≥ 1 x ≥ – 1 Example 4

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Method 1 Use algebra to solve the inequality. Step 2 Consider the radicand. The radicand cannot be negative. Solve for x. x + 2 ≥ 1 x ≥ –1 The solution of is x ≥ – 1. Example 4

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Method 1 Use a graph and a table. 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. Example 4

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Solve Practice

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Sources Holt Algebra 2 Glencoe Algebra 2

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