1. Solving Radical Equations and Inequalities
Section 5.4 beginning on page 262

2. Solving Radical Equations
Step 1: Isolate the radical on one side.
Step 2: Raise each side to the same power to eliminate the root.
Step 3: Solve the resulting equation.
Step 4: Check your answers for extraneous solutions.
Example 1: Solve
a) 2 𝑥+1 =4 2
𝑥+1 =2
( 𝑥+1 ) 2 = 2 2
𝑥+1=4
2 𝑥+1 =4
𝑥=3 =4
2 4 =4
2(2)=4

3. Solving Radical Equations
Step 1: Isolate the radical on one side.
Step 2: Raise each side to the same power to eliminate the root.
Step 3: Solve the resulting equation.
Step 4: Check your answers for extraneous solutions.
Example 1: Solve
b) 3 2𝑥−9 −1=2
3 2𝑥−9 =3
( 3 2𝑥−9 ) 3 = 3 3
2𝑥−9=27
2𝑥=36
3 2(18)−9 −1=2
𝑥=18
3 36−9 −1=2
3 27 −1=2
3−1=2

4. Extraneous Solutions Example 3: Solve 𝑥+1= 7𝑥+15 𝑥+1= 7𝑥+15 𝑥+1= 7𝑥+15
𝑥+1= 7𝑥+15
𝑥+1= 7𝑥+15
𝑥+1= 7𝑥+15
7+1= 7(7)+15
−2+1= 7(−2)+15
(𝑥+1) 2 = ( 7𝑥+15 ) 2 8=
−1= −14+15
𝑥 2 +2𝑥+1=7𝑥+15
8= 64
−1≠ 1
𝑥 2 −5𝑥−14=0
Since the radical is in the original problem, we are only looking for the principal (positive) root.
𝑥−7 𝑥+2 =0
𝑥=7
𝑥=7, 𝑥=−2

5. Solving Equations With Two Radicals
Example 4: Solve
𝑥+2 +1= 3−𝑥
𝑥+2 =−𝑥
( 𝑥+2 ) 2 = (−𝑥) 2
( 𝑥+2 +1) 2 = ( 3−𝑥 ) 2
𝑥+2+2 𝑥+2 +1=3−𝑥
𝑥+2= 𝑥 2
0= 𝑥 2 −𝑥−2
𝑥+2 𝑥+2 +3=3−𝑥
0=(𝑥−2)(𝑥+1)
2 𝑥+2 =−2𝑥 2
𝑥=2,𝑥=−1
𝑥=−1
Check:
(2)+2 +1= 3−2
(−1)+2 +1= 3−(−1)
4 +1= 1
1 +1= 4
2+1≠1
1+1=2

6. Solving an Equation with a Rational Exponent
Example 5: Solve
Step 1: Isolate what is being raised to a power.
Step 2 : Raise each side to the reciprocal power.
Step 3: Solve the resulting equation.
Step 4: Check solution(s).
2𝑥 =10
2𝑥 =8
2𝑥 =
2𝑥= ( 3 8 ) 4
2𝑥= (2) 4
2𝑥=16
2(8) =10
𝑥=8
=10
𝑥=8
2 3 +2=10
8+2=10

7. Solving a Radical Inequality
** The same rules apply when we solve inequalities except:
If you multiply or divide both sides by a negative, flip the inequality.
An even root can not be less than 0.
We must consider the possible values of the radicand
We now know that 𝑥≤17 but there are values less than 17 that will not work here (due to the even root), so we need to find out what will work.
Example 7: Solve
3 𝑥−1 ≤12 3
𝑥−1≥0
𝑥≥1
𝑥−1 ≤4
𝑥−1 2 ≤ 4 2
1≤𝑥≤17
𝑥−1≤16
𝑥≤17

8. Monitoring Progress Solve the equation. Check your solution(s).
1) 3 𝑥 −9=−6 2) 𝑥+25 =2 3) 2 3 𝑥−3 =4
5) 10𝑥+9 =𝑥+3 6) 2𝑥+5 = 𝑥+7 7) 𝑥+6 −2= 𝑥−2
8) 3𝑥 =−3 9) 𝑥 =𝑥 10) 𝑥 =8
11) Solve (a) 2 𝑥 −3≥3 and (b) 4 3 𝑥+1 <8
𝑥=27
𝑥=−21
𝑥=11
𝑥=27 𝑎𝑛𝑑 𝑥=0
𝑥=2
𝑥=3
𝑥=−9
𝑥=3
𝑥=14
𝑥≥9
𝑥<7