Solving Radical Equations and Inequalities 5-8 Warm Up Lesson Presentation Lesson Quiz Holt McDougal Algebra 2 Holt Algebra 2

Warm Up Simplify each expression. Assume all variables are positive. Write each expression in radical form. 1. 2. 4. 3. 5. 6.

Objective Solve radical equations and inequalities.

Vocabulary radical equation radical inequality

A radical equation contains a variable within a radical 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.

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

Example 1A: Solving Equations Containing One Radical Solve each equation. Check Subtract 5. Simplify. Square both sides.  Simplify. Solve for x.

Example 1B: Solving Equations Containing One Radical Solve each equation. Check 3 7 7 5x - 7 84 = Divide by 7. 7 Simplify. Cube both sides.  Simplify. Solve for x.

Check It Out! Example 1a Solve the equation. Check Subtract 4. Simplify.  Square both sides. Simplify. Solve for x.

Check It Out! Example 1b Solve the equation. Check Cube both sides. Simplify. Solve for x. 

Check It Out! Example 1c Solve the equation. Check Divide by 6. Square both sides. Simplify.  Solve for x.

Example 2: Solving Equations Containing Two Radicals Solve Square both sides. 7x + 2 = 9(3x – 2) Simplify. 7x + 2 = 27x – 18 Distribute. 20 = 20x Solve for x. 1 = x

Example 2 Continued Check 3 3 

Check It Out! Example 2a Solve each equation. Square both sides. 8x + 6 = 9x Simplify. 6 = x Solve for x. Check 

Check It Out! Example 2b Solve each equation. Cube both sides. x + 6 = 8(x – 1) Simplify. x + 6 = 8x – 8 Distribute. 14 = 7x Solve for x. 2 = x Check 2 2 

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

Example 3 Continued Method 2 Use algebra to solve the equation. Step 1 Solve for x. Square both sides. –3x + 33 = 25 – 10x + x2 Simplify. 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 x 6 6  Because x = 8 is extraneous, the only solution is x = –1.

Check It Out! Example 3a Continued Method 2 Use algebra to solve the equation. Step 1 Solve for x. Square both sides. Simplify. 2x + 14 = x2 + 6x + 9 0 = x2 + 4x – 5 Write in standard form. Factor. 0 = (x + 5)(x – 1) x + 5 = 0 or x – 1 = 0 Solve for x. x = –5 or x = 1

Check It Out! Example 3a Continued Method 1 Use algebra to solve the equation. Step 2 Use substitution to check for extraneous solutions. 2 –2 x 4 4  Because x = –5 is extraneous, the only solution is x = 1.

Check It Out! Example 3b Continued Method 2 Use algebra to solve the equation. Step 1 Solve for x. Square both sides. Simplify. –9x + 28 = x2 – 8x + 16 0 = x2 + x – 12 Write in standard form. Factor. 0 = (x + 4)(x – 3) x + 4 = 0 or x – 3 = 0 Solve for x. x = –4 or x = 3

Check It Out! Example 3b Continued Method 1 Use algebra to solve the equation. Step 2 Use substitution to check for extraneous solutions.  

To find a power, multiply the exponents. Remember!

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

Example 4B: Solving Equations with Rational Exponents 2x = (4x + 8) 1 2 Step 1 Solve for x. Raise both sides to the reciprocal power. (2x)2 = [(4x + 8) ]2 1 2 4x2 = 4x + 8 Simplify. 4x2 – 4x – 8 = 0 Write in standard form. 4(x2 – x – 2) = 0 Factor out the GCF, 4. 4(x – 2)(x + 1) = 0 Factor. 4 ≠ 0, x – 2 = 0 or x + 1 = 0 Solve for x. x = 2 or x = –1

Step 2 Use substitution to check for extraneous solutions. Example 4B Continued Step 2 Use substitution to check for extraneous solutions. 2x = (4x + 8) 1 2 2(2) (4(2) + 8) 4 16 4 4  2x = (4x + 8) 1 2 2(–1) (4(–1) + 8) –2 4 –2 2 x The only solution is x = 2.

Check It Out! Example 4a Solve each equation. (x + 5) = 3 1 3 (x + 5) = 3 Write in radical form. Cube both sides. x + 5 = 27 Simplify. x = 22 Solve for x.

Raise both sides to the reciprocal power. Check It Out! Example 4b 1 2 (2x + 15) = x Step 1 Solve for x. [(2x + 15) ]2 = (x)2 1 2 Raise both sides to the reciprocal power. 2x + 15 = x2 Simplify. x2 – 2x – 15 = 0 Write in standard form. (x – 5)(x + 3) = 0 Factor. x – 5 = 0 or x + 3 = 0 Solve for x. x = 5 or x = –3

Check It Out! Example 4b Continued Step 2 Use substitution to check for extraneous solutions. (2(5) + 15) 5 1 2 (2x + 15) = x (10 + 15) 5  5 5 (2(–3) + 15) –3 1 2 (–6 + 15) –3 x (2x + 15) = x 3 –3 The only solution is x = 5.

Raise both sides to the reciprocal power. [3(x + 6) ]2 = (9)2 Check It Out! Example 4c 1 2 3(x + 6) = 9 Raise both sides to the reciprocal power. [3(x + 6) ]2 = (9)2 1 2 9(x + 6) = 81 Simplify. 9x + 54 = 81 Distribute 9. 9x = 27 Simplify. x = 3 Solve for x.

Example 5 Continued Method 2 Use algebra to solve the inequality. Step 1 Solve for x. Subtract 3. Square both sides. Simplify. 2x ≤ 42 Solve for x. x ≤ 21

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

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

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

Check It Out! Example 5b Continued Method 2 Use algebra to solve the inequality. Step 1 Solve for x. Cube both sides. x + 2 ≥ 1 Solve for x. x ≥ –1

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

Lesson Quiz: Part I Solve each equation or inequality. 1. x = 16 2. x = –5 3. x = 2 4. x = 14 5. x ≥ 59