Problem 42 - Intermediate Algebra

Question

Solve. \(\sqrt[3]{-4 x-3}=\sqrt[3]{-x-15}\)

Step-by-Step Solution

Answer

The solution to the equation is \(x = 4\).

Step 1: Understand the Problem

We have two cube roots on each side of the equation: \(\sqrt[3]{-4x-3}\) and \(\sqrt[3]{-x-15}\). We need to find the value of \(x\) that makes these two expressions equal.

Step 2: Set the Radicands Equal

Since the cube roots are equal, their radicands must be equal as well. Therefore, we set the expressions inside the cube roots equal to each other: \(-4x - 3 = -x - 15\).

Step 3: Solve for x

To find \(x\), we'll start by isolating \(x\) on one side. Add \(4x\) to both sides of the equation to get \(-3 = 3x - 15\).

Step 4: Simplify Further

Add 15 to both sides to isolate the term with \(x\): -3 + 15 = 3x, which simplifies to \(12 = 3x\).

Step 5: Solve for x

Divide both sides by 3 to solve for \(x\): \(x = \frac{12}{3}\). Therefore, \(x = 4\).

Step 6: Verify the Solution

Substitute \(x = 4\) back into the original expressions to verify: \(\sqrt[3]{-4(4) - 3} = \sqrt[3]{-x - 15}\) simplifies to \(\sqrt[3]{-19} = \sqrt[3]{-19}\). Since both sides are equal, our solution is correct.

Key Concepts

Understanding Radicands

When solving equations involving cube roots, it's essential to grasp the concept of radicands. A radicand is the term located under the radical sign. In the equation \(\sqrt[3]{-4x-3}=\sqrt[3]{-x-15}\), both \(-4x-3\) and \(-x-15\) are radicands. Cube roots, like square roots, require that both sides of the equation, under the radical, be equal for the equality to hold.
Therefore, by setting the radicands equal to each other, as in \(-4x-3=-x-15\), we eliminate the radicals, simplifying the problem into an easier algebraic form.

Key points to remember about radicands in cube root equations are:

• For the cube roots to be equal, the radicands must be equal—making it easier to solve the equation by dealing with regular algebraic terms instead of radicals.
• Unlike square roots, negative numbers are entirely permissible under cube roots.

Mastering radicands elevates your solving capability as it allows a crucial step in simplifying complex root equations.

Isolation of Variables

Once the radicands are set equal, the next crucial step is the isolation of variables. The goal is to rearrange the equation to have all terms involving the variable \(x\) on one side.
In our example, the equation is \(-4x-3=-x-15\). We begin isolating \(x\) by adding \(4x\) to both sides. This operation transforms the equation into \(-3 = 3x - 15\).
Next, to further isolate \(x\), we add 15 to both sides, resulting in \(12 = 3x\). Finally, divide both sides by 3 to solve for \(x\), giving \(x = 4\).

• This process systematically reduces the problem into a simpler, one-variable equation.
• Careful balancing of both sides is key, reflecting algebraic principles of equal transformations.

Effectively isolating variables is a vital skill, facilitating solution derivation in a structured and clear manner.

Verification of Solutions

After finding \(x\), verifying our solution is obligatory to ensure no mistakes were made. Verification involves substituting the solution back into the initial equation.
In our case, we substitute \(x=4\) back into the original cube root expressions as: \(\sqrt[3]{-4(4) - 3}\) and \(\sqrt[3]{-4 - 15}\). This simplifies to \(\sqrt[3]{-19}\) on both sides.

Verification is crucial for several reasons:

• Confirms the correctness of the solution by ensuring the expressions still satisfy the original equation.
• Counters missteps made in calculation or simplification stages of solving.
• Solidifies understanding by revisiting each step, reinforcing algebraic concepts.

Thus, diligent verification offers confidence in your solutions, cementing each step's accuracy throughout the problem-solving process.