Let's solve the equation [tex]\( \sqrt[3]{d-4} = \sqrt[3]{3-2d} \)[/tex] for [tex]\( d \)[/tex].
### Step 1: Cube both sides
To eliminate the cube roots, we will cube both sides of the equation.
[tex]\[ \left( \sqrt[3]{d-4} \right)^3 = \left( \sqrt[3]{3-2d} \right)^3 \][/tex]
This simplifies to:
[tex]\[ d - 4 = 3 - 2d \][/tex]
### Step 2: Rearrange the equation
Next, we need to gather all terms involving [tex]\( d \)[/tex] on one side of the equation. Add [tex]\( 2d \)[/tex] to both sides:
[tex]\[ d - 4 + 2d = 3 \][/tex]
Combine like terms:
[tex]\[ 3d - 4 = 3 \][/tex]
### Step 3: Solve for [tex]\( d \)[/tex]
Now, isolate [tex]\( d \)[/tex]. Add 4 to both sides of the equation:
[tex]\[ 3d = 7 \][/tex]
Divide both sides by 3:
[tex]\[ d = \frac{7}{3} \][/tex]
This is the solution.
### Step 4: Verify the solution
To ensure that [tex]\( d = \frac{7}{3} \)[/tex] is a valid solution, substitute it back into the original equation:
[tex]\[ \sqrt[3]{\frac{7}{3} - 4} = \sqrt[3]{3 - 2 \left(\frac{7}{3}\right)} \][/tex]
Simplify inside the cube roots:
[tex]\[ \sqrt[3]{\frac{7}{3} - \frac{12}{3}} = \sqrt[3]{3 - \frac{14}{3}} \][/tex]
[tex]\[ \sqrt[3]{\frac{7-12}{3}} = \sqrt[3]{\frac{9-14}{3}} \][/tex]
[tex]\[ \sqrt[3]{\frac{-5}{3}} = \sqrt[3]{\frac{-5}{3}} \][/tex]
Both sides are equal, confirming that [tex]\( d = \frac{7}{3} \)[/tex] is indeed a solution.
Thus, the solution to the equation is:
[tex]\[ d = \frac{7}{3} \][/tex]
