Problem 47 - Algebra for College Students

Question

Solve each equation. Don't forget to check each of your potential solutions. \(\sqrt[3]{2 x+5}=\sqrt[3]{4-x}\)

Step-by-Step Solution

Answer

The solution is \(x = -\frac{1}{3}\).

Step 1: Analyze the Equation

The given equation is \(\sqrt[3]{2x + 5} = \sqrt[3]{4 - x}\). We need to consider solving for \(x\) by removing the cube roots.

Step 2: Remove the Cube Roots

Cube both sides to eliminate the cube roots: \[(\sqrt[3]{2x + 5})^3 = (\sqrt[3]{4 - x})^3\]Simplifying this gives \[2x + 5 = 4 - x.\]

Step 3: Isolate the Variable

Add \(x\) to both sides to get:\[2x + x + 5 = 4\]This simplifies to \[3x + 5 = 4.\]

Step 4: Solve for x

Subtract 5 from both sides:\[3x = 4 - 5\]This simplifies to\[3x = -1.\]Now, divide by 3:\[x = -\frac{1}{3}\]

Step 5: Verify the Solution

Substitute \(x = -\frac{1}{3}\) back into the original equation:\[\sqrt[3]{2(-\frac{1}{3}) + 5} = \sqrt[3]{4 - (-\frac{1}{3})}\]Calculating the left side, we have:\[\sqrt[3]{-\frac{2}{3} + 5} = \sqrt[3]{\frac{13}{3}}\]Calculating the right side, we have:\[\sqrt[3]{4 + \frac{1}{3}} = \sqrt[3]{\frac{13}{3}}\]Since both sides equal \(\sqrt[3]{\frac{13}{3}}\), \(x = -\frac{1}{3}\) is a valid solution.

Key Concepts

Cube Roots

Understanding cube roots is essential when solving equations like the one given in the exercise. The cube root of a number is a value that, when multiplied by itself three times, returns the original number. For instance, the cube root of 8 is 2 since 2 × 2 × 2 = 8. In general, the cube root is denoted by \ \(\sqrt[3]{number}\ \).

In the equation \ \(\sqrt[3]{2x + 5} = \sqrt[3]{4 - x}\ \), we can see cube roots on both sides. To eliminate them, we cube both sides of the equation. Cubing the cube roots effectively "undoes" the root, simplifying to the expressions inside:

• \ \( (\sqrt[3]{2x + 5})^3 = 2x + 5\ \)
• \ \((\sqrt[3]{4 - x})^3 = 4 - x\ \)

This step clears the equation of cube roots, making it simpler to solve for \(x\). Understanding how to handle cube roots is critical for manipulating and solving such equations.

Equation Verification

Verifying a solution to an equation ensures that your solution is correct and satisfies the original equation. After finding a potential solution for \(x\), it is crucial to substitute it back into the original equation to check if both sides are equal.

For the equation \ \(\sqrt[3]{2x + 5} = \sqrt[3]{4 - x}\ \), after finding \(x = -\frac{1}{3}\), we substitute \(-\frac{1}{3}\) back into the equation:

• Substitute into the left side: \ \(\sqrt[3]{2(-\frac{1}{3}) + 5} = \sqrt[3]{-\frac{2}{3} + 5} = \sqrt[3]{\frac{13}{3}}\ \)
• Substitute into the right side: \ \(\sqrt[3]{4 -(-\frac{1}{3})} = \sqrt[3]{4 + \frac{1}{3}} = \sqrt[3]{\frac{13}{3}}\ \)

Both sides simplify to \ \(\sqrt[3]{\frac{13}{3}}\ \), confirming that \(x = -\frac{1}{3}\) is indeed a valid solution. Never skip this verification step, as it confirms your answer's accuracy.

Variable Isolation

Variable isolation is a fundamental algebraic technique used to find the value of an unknown variable. In the equation \ \(2x + 5 = 4 - x\ \), the first step to isolate \(x\) is to get all terms involving \(x\) on one side of the equation.

Here's a breakdown of the steps to isolate \(x\):

• Move \(-x\) to the left side by adding \(x\) to both sides: \ \(2x + x + 5 = 4\ \), simplifying to \ \(3x + 5 = 4\ \)
• Subtract 5 from both sides to further isolate \(x\): \ \(3x = 4 - 5\ \), which simplifies to \ \(3x = -1\ \)
• Finally, divide both sides by 3 to solve for \(x\): \ \(x = -\frac{1}{3}\ \)

These steps transform the original equation into a much simpler form, allowing us to solve for \(x\). Mastering variable isolation is essential to tackle complex algebraic equations successfully.