Solve the Rational Equation for x square root of 3x+4=1+ square root of 3x-11

$\sqrt{3 x + 4} = 1 + \sqrt{3 x - 11}$

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{3 x + 4}}^{2} = {(1 + \sqrt{3 x - 11})}^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 x + 4}$ as ${(3 x + 4)}^{\frac{1}{2}}$ .

${({(3 x + 4)}^{\frac{1}{2}})}^{2} = {(1 + \sqrt{3 x - 11})}^{2}$

Simplify the left side.

Simplify ${({(3 x + 4)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(3 x + 4)}^{\frac{1}{2}})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(3 x + 4)}^{\frac{1}{2} \cdot 2} = {(1 + \sqrt{3 x - 11})}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(3 x + 4)}^{\frac{1}{2} \cdot 2} = {(1 + \sqrt{3 x - 11})}^{2}$

Rewrite the expression.

${(3 x + 4)}^{1} = {(1 + \sqrt{3 x - 11})}^{2}$

Simplify.

$3 x + 4 = {(1 + \sqrt{3 x - 11})}^{2}$

Simplify the right side.

Simplify ${(1 + \sqrt{3 x - 11})}^{2}$ .

Rewrite ${(1 + \sqrt{3 x - 11})}^{2}$ as $(1 + \sqrt{3 x - 11}) (1 + \sqrt{3 x - 11})$ .

$3 x + 4 = (1 + \sqrt{3 x - 11}) (1 + \sqrt{3 x - 11})$

Expand $(1 + \sqrt{3 x - 11}) (1 + \sqrt{3 x - 11})$ using the FOIL Method.

Apply the distributive property.

$3 x + 4 = 1 (1 + \sqrt{3 x - 11}) + \sqrt{3 x - 11} (1 + \sqrt{3 x - 11})$

Apply the distributive property.

$3 x + 4 = 1 \cdot 1 + 1 \sqrt{3 x - 11} + \sqrt{3 x - 11} (1 + \sqrt{3 x - 11})$

Apply the distributive property.

$3 x + 4 = 1 \cdot 1 + 1 \sqrt{3 x - 11} + \sqrt{3 x - 11} \cdot 1 + \sqrt{3 x - 11} \sqrt{3 x - 11}$

Simplify and combine like terms.

Simplify each term.

Multiply $1$ by $1$ .

$3 x + 4 = 1 + 1 \sqrt{3 x - 11} + \sqrt{3 x - 11} \cdot 1 + \sqrt{3 x - 11} \sqrt{3 x - 11}$

Multiply $\sqrt{3 x - 11}$ by $1$ .

$3 x + 4 = 1 + \sqrt{3 x - 11} + \sqrt{3 x - 11} \cdot 1 + \sqrt{3 x - 11} \sqrt{3 x - 11}$

Multiply $\sqrt{3 x - 11}$ by $1$ .

$3 x + 4 = 1 + \sqrt{3 x - 11} + \sqrt{3 x - 11} + \sqrt{3 x - 11} \sqrt{3 x - 11}$

Multiply $\sqrt{3 x - 11} \sqrt{3 x - 11}$ .

Raise $\sqrt{3 x - 11}$ to the power of $1$ .

$3 x + 4 = 1 + \sqrt{3 x - 11} + \sqrt{3 x - 11} + {\sqrt{3 x - 11}}^{1} \sqrt{3 x - 11}$

Raise $\sqrt{3 x - 11}$ to the power of $1$ .

$3 x + 4 = 1 + \sqrt{3 x - 11} + \sqrt{3 x - 11} + {\sqrt{3 x - 11}}^{1} {\sqrt{3 x - 11}}^{1}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$3 x + 4 = 1 + \sqrt{3 x - 11} + \sqrt{3 x - 11} + {\sqrt{3 x - 11}}^{1 + 1}$

Add $1$ and $1$ .

$3 x + 4 = 1 + \sqrt{3 x - 11} + \sqrt{3 x - 11} + {\sqrt{3 x - 11}}^{2}$

Rewrite ${\sqrt{3 x - 11}}^{2}$ as $3 x - 11$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 x - 11}$ as ${(3 x - 11)}^{\frac{1}{2}}$ .

$3 x + 4 = 1 + \sqrt{3 x - 11} + \sqrt{3 x - 11} + {({(3 x - 11)}^{\frac{1}{2}})}^{2}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$3 x + 4 = 1 + \sqrt{3 x - 11} + \sqrt{3 x - 11} + {(3 x - 11)}^{\frac{1}{2} \cdot 2}$

Combine $\frac{1}{2}$ and $2$ .

$3 x + 4 = 1 + \sqrt{3 x - 11} + \sqrt{3 x - 11} + {(3 x - 11)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$3 x + 4 = 1 + \sqrt{3 x - 11} + \sqrt{3 x - 11} + {(3 x - 11)}^{\frac{2}{2}}$

Rewrite the expression.

$3 x + 4 = 1 + \sqrt{3 x - 11} + \sqrt{3 x - 11} + {(3 x - 11)}^{1}$

Simplify.

$3 x + 4 = 1 + \sqrt{3 x - 11} + \sqrt{3 x - 11} + 3 x - 11$

Subtract $11$ from $1$ .

$3 x + 4 = - 10 + \sqrt{3 x - 11} + \sqrt{3 x - 11} + 3 x$

Add $\sqrt{3 x - 11}$ and $\sqrt{3 x - 11}$ .

$3 x + 4 = - 10 + 2 \sqrt{3 x - 11} + 3 x$

Solve for $2 \sqrt{3 x - 11}$ .

Rewrite the equation as $- 10 + 2 \sqrt{3 x - 11} + 3 x = 3 x + 4$ .

$- 10 + 2 \sqrt{3 x - 11} + 3 x = 3 x + 4$

Move all terms not containing $2 \sqrt{3 x - 11}$ to the right side of the equation.

Add $10$ to both sides of the equation.

$2 \sqrt{3 x - 11} + 3 x = 3 x + 4 + 10$

Subtract $3 x$ from both sides of the equation.

$2 \sqrt{3 x - 11} = 3 x + 4 + 10 - 3 x$

Combine the opposite terms in $3 x + 4 + 10 - 3 x$ .

Subtract $3 x$ from $3 x$ .

$2 \sqrt{3 x - 11} = 0 + 4 + 10$

Add $0$ and $4$ .

$2 \sqrt{3 x - 11} = 4 + 10$

Add $4$ and $10$ .

$2 \sqrt{3 x - 11} = 14$

To remove the radical on the left side of the equation, square both sides of the equation.

${(2 \sqrt{3 x - 11})}^{2} = 14^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 x - 11}$ as ${(3 x - 11)}^{\frac{1}{2}}$ .

${(2 {(3 x - 11)}^{\frac{1}{2}})}^{2} = 14^{2}$

Simplify the left side.

Simplify ${(2 {(3 x - 11)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $2 {(3 x - 11)}^{\frac{1}{2}}$ .

$2^{2} {({(3 x - 11)}^{\frac{1}{2}})}^{2} = 14^{2}$

Raise $2$ to the power of $2$ .

$4 {({(3 x - 11)}^{\frac{1}{2}})}^{2} = 14^{2}$

Multiply the exponents in ${({(3 x - 11)}^{\frac{1}{2}})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$4 {(3 x - 11)}^{\frac{1}{2} \cdot 2} = 14^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$4 {(3 x - 11)}^{\frac{1}{2} \cdot 2} = 14^{2}$

Rewrite the expression.

$4 {(3 x - 11)}^{1} = 14^{2}$

Simplify.

$4 (3 x - 11) = 14^{2}$

Apply the distributive property.

$4 (3 x) + 4 \cdot - 11 = 14^{2}$

Multiply.

Multiply $3$ by $4$ .

$12 x + 4 \cdot - 11 = 14^{2}$

Multiply $4$ by $- 11$ .

$12 x - 44 = 14^{2}$

Simplify the right side.

Raise $14$ to the power of $2$ .

$12 x - 44 = 196$

Solve for $x$ .

Move all terms not containing $x$ to the right side of the equation.

Add $44$ to both sides of the equation.

$12 x = 196 + 44$

Add $196$ and $44$ .

$12 x = 240$

Divide each term in $12 x = 240$ by $12$ and simplify.

Divide each term in $12 x = 240$ by $12$ .

$\frac{12 x}{12} = \frac{240}{12}$

Simplify the left side.

Cancel the common factor of $12$ .

Cancel the common factor.

$\frac{12 x}{12} = \frac{240}{12}$

Divide $x$ by $1$ .

$x = \frac{240}{12}$

Simplify the right side.

Divide $240$ by $12$ .

$x = 20$

Solve the Rational Equation for x square root of 3x+4=1+ square root of 3x-11

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