Warm up 1. Solve 2. Solve 3. Decompose to partial fractions -1/2, 1 -1, -2 x<-2 or -1<x<5 Warm up

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

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

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

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

Example 2 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. Example 2

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

Solve -75 Practice

Solve: 8, (24 does not work in the original equation) Practice

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!

Example 3 Solve . Method 1 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 Example 3

Example 3 Method 1 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. Example 3

Example 3 Solve . 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. Example 3

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

Example 4 Method 1 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. Example 4

Example 4 Solve . Method 1 Use a graph and a table. 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. The solution is x ≥ –1. Example 4

Solve x>7/2 Practice

Holt Algebra 2 Glencoe Algebra 2 Sources