Solving Radical Equations and Inequalities Section 5.4 beginning on page 262

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 2 3+1 =4 2 4 =4 2(2)=4

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

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= 49+15 −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

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

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 3 4 +2=10 2𝑥 3 4 =8 2𝑥 3 4 4 3 = 8 4 3 2𝑥= ( 3 8 ) 4 2𝑥= (2) 4 2𝑥=16 2(8) 3 4 +2=10 𝑥=8 16 3 4 +2=10 𝑥=8 2 3 +2=10 8+2=10

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

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 1 3 =−3 9) 𝑥+6 1 2 =𝑥 10) 𝑥+2 3 4 =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