Warm up

• 1. Solve

• 2. Solve

• 3. Decompose to partial fractions

2

3

1

1

12

1

tt

32

12

x

xx

20

752

xx

x

Lesson 4-7 Radical Equations and Inequalities

Objective: To solve radical equations and inequalities

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.

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

Remember!

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

Solve Radical Equations

Original equation

Answer: The solution checks. The solution is 38.

Check

Simplify.

Replace y with 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

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 = x2 + 6x + 9

0 = x2 + 4x – 5

0 = (x + 5)(x – 1)

x + 5 = 0 or x – 1 = 0

x = –5 or x = 1

Example 2

Method 1 Use algebra to solve the equation.

Step 2 Use substitution to check for extraneous solutions.

4 4 x

Because x = –5 is extraneous, the only solution is x = 1.

2 14 35 5

2 –2

Example 2

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

Practice

• Solve 3 1137 x

• Solve:

Practice

12211 xx

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!

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 .

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

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

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

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

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

• Solve

Practice

456 x

Sources

• Holt Algebra 2• Glencoe Algebra 2