Chapter 6.7, Problem 24PPS

To determine

To Solve:

  $6+\sqrt{3x+1}=11$

Answer to Problem 24PPS

x = 8

Explanation of Solution

Given Information:

Given equation:

  $6+\sqrt{3x+1}=11$

Concept Used :

Solve Radical equations:

• Isolate the radical on one side of the equation.
• Raise each side of the equation to a power equal to the index of the radical to eliminate the radical .
• Solve the resulting polynomial equation.Check your results.
• When solving radical equations, the result may be a number that does not satisfy the original equation . Such a number is called an extraneous solution.

Calculation:

• Given $6+\sqrt{3x+1}=11$

  $\begin{array}{l}6+\sqrt{3x+1}=11\\ \Rightarrow 6+\sqrt{3x+1}-6=11-\mathrm{6.............}[\text{Subtract 6 from each side]}\\ \Rightarrow \sqrt{3x+1}=11-6=5\\ \Rightarrow {\left(\sqrt{3x+1}\right)}^2=5^2\mathrm{......................}[\text{Squaring each side]}\\ \Rightarrow 3x+1=\mathrm{25.............................}[Simplify]\\ \Rightarrow 3x+1-1=25-\mathrm{1..................}[\text{Subtract 1 from each side]}\\ \Rightarrow 3x=24\\ \Rightarrow \frac{3x}{3}=\frac{24}{3}\mathrm{...............................}[\text{Divide each side by 3]}\\ \Rightarrow x=8\\ \text{Check:}\\ \text{Substitute the value in the left hand side of the given equation:}\\ 6+\sqrt{3x+1}=11\\ \text{Left hand side = }6+\sqrt{3(8)+1}=6+\sqrt{24+1}=6+5=11=\text{Right hand side}\\ \text{So, the solution is }x=8\end{array}$

Check :