Questions: Solve the cube root equation. Check for extraneous solutions. sqrt[3]3x-1 - 6 = -4 x=-3 x=3 x=7/3 x=1

Solve the cube root equation. Check for extraneous solutions. sqrt[3]3x-1 - 6 = -4 x=-3 x=3 x=7/3 x=1
Transcript text: Solve the cube root equation. Check for extraneous solutions. \[ \sqrt[3]{3 x-1}-6=-4 \] $x=-3$ $x=3$ $x=\frac{7}{3}$ $x=1$
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Solution

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Solution Steps

To solve the cube root equation, first isolate the cube root term. Then, cube both sides to eliminate the cube root. Solve the resulting equation for \( x \). Finally, check each solution to ensure it is not extraneous by substituting back into the original equation.

Step 1: Isolate the Cube Root Term

Start with the equation: \[ \sqrt[3]{3x - 1} - 6 = -4 \] Add 6 to both sides to isolate the cube root: \[ \sqrt[3]{3x - 1} = 2 \]

Step 2: Eliminate the Cube Root

Cube both sides to remove the cube root: \[ 3x - 1 = 2^3 \] \[ 3x - 1 = 8 \]

Step 3: Solve for \( x \)

Add 1 to both sides: \[ 3x = 9 \] Divide by 3: \[ x = 3 \]

Step 4: Check for Extraneous Solutions

Substitute \( x = 3 \) back into the original equation to verify: \[ \sqrt[3]{3(3) - 1} - 6 = -4 \] \[ \sqrt[3]{9 - 1} - 6 = -4 \] \[ \sqrt[3]{8} - 6 = -4 \] \[ 2 - 6 = -4 \] The solution satisfies the original equation.

Final Answer

The solution is \(\boxed{x = 3}\).

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