Question

solve for x, where x is a real number. (sqrt{3x + 14}-sqrt{7x - 14}=0). if there is more than one solution, separate them with commas. if there is no solution, click on no solution. x =

Explanation:

Step1: Isolate one of the square - roots

Add $\sqrt{7x - 14}$ to both sides of the equation $\sqrt{3x + 14}-\sqrt{7x - 14}=0$. We get $\sqrt{3x + 14}=\sqrt{7x - 14}$.

Step2: Square both sides

Squaring both sides of the equation $\sqrt{3x + 14}=\sqrt{7x - 14}$, we have $3x + 14=7x - 14$ (since if $\sqrt{a}=\sqrt{b}$, then $a = b$ for $a\geq0$ and $b\geq0$).

Step3: Solve the resulting linear equation

Subtract $3x$ from both sides: $14 = 7x-3x - 14$, which simplifies to $14 = 4x-14$. Then add 14 to both sides: $14 + 14=4x$, so $28 = 4x$. Divide both sides by 4, we get $x = 7$.

Step4: Check for extraneous solutions

For the original equation $\sqrt{3x + 14}-\sqrt{7x - 14}=0$, when $x = 7$, $\sqrt{3\times7+14}-\sqrt{7\times7 - 14}=\sqrt{21 + 14}-\sqrt{49 - 14}=\sqrt{35}-\sqrt{35}=0$.

Answer:

$7$