• 1. Solve
• 2. Solve
1
1
3


2t  1 t  1 2
x  x 1
3
x2
2
• 3. Decompose to partial fractions
Warm up
5x  7
x 2  x  20
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.
Remember!
For a square root, the index of the radical is 2.
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
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.
Helpful Hint
You can use the intersect feature on a graphing
calculator to find the point where the two curves
intersect.
Method 1 Use algebra to solve the equation.
Step 1 Solve for x.
Square both sides.
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
Simplify.
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
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
3
7

3

x  11
• Solve
Practice
• Solve:
Practice
x  1  1  2 x  12
A radical inequality is an inequality that
contains a variable within a radical. You can
solve radical inequalities by graphing or using
algebra.
Remember!
A radical expression with an even index and a
negative radicand has no real roots.
Solve
.
Method 1 Use algebra to solve the inequality.
Step 1 Solve for x.
Subtract 2.
Square both sides.
Simplify.
x–3≤9
x ≤ 12
Example 3
Solve for x.
Method 1 Use algebra to solve the inequality.
Step 2 Consider the radicand.
x–3≥0
The radicand cannot be negative.
x≥3
The solution of
12, or 3 ≤ x ≤ 12.
Example 3
Solve for x.
is x ≥ 3 and x ≤
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
Method 1 Use algebra to solve the inequality.
Step 1 Solve for x.
Cube both sides.
x+2≥1
x ≥ –1
Example 4
Solve for x.
Method 1 Use algebra to solve the inequality.
Step 2 Consider the radicand.
x+2≥1
x ≥ –1
The solution of
Example 4
The radicand cannot be negative.
Solve for x.
is x ≥ –1.
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
Practice
6x  5  4
• Holt Algebra 2
• Glencoe Algebra 2
Sources