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

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

3. 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.

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

5. 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

6. 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

7. 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

8. 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

9. Example 2 Method 1 Use algebra to solve the equation.
Step 2 Use substitution to check for extraneous solutions. –2 x 
Because x = –5 is extraneous, the only solution is x = 1.
Example 2

10. 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

11. Solve
-75
Practice

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

13. 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!

14. 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

15. 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

16. 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

17. 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

18. 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

19. 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

20. Solve
x>7/2
Practice

21. Holt Algebra 2
Glencoe Algebra 2
Sources