Problem 41 - Intermediate Algebra

Question

Solve. \(\sqrt[3]{-6 x-1}=\sqrt[3]{-2 x-5}\)

Step-by-Step Solution

Answer

The solution is \(x = 1\).

Step 1: Equate the Radicals

Since both sides of the equation have cube roots, we can equate the expressions inside the radicals directly because if \( \sqrt[3]{a} = \sqrt[3]{b} \), then \( a = b \). Thus, set \(-6x - 1 = -2x - 5\).

Step 2: Rearrange the Equation

Move all terms involving \(x\) to one side and constant terms to the other side. Add \(2x\) to both sides to get \(-6x + 2x = -5 + 1\). This simplifies to \(-4x = -4\).

Step 3: Solve for \(x\)

Divide both sides by \(-4\) to solve for \(x\). This gives \(x = 1\).

Key Concepts

Cube Roots

Cube roots are an important concept in mathematics, particularly when working with radical equations. A cube root of a number is a value that, when multiplied by itself twice, gives the original number. For example, the cube root of 8 is 2 because \(2 \times 2 \times 2 = 8\).

When dealing with cube roots in equations, we often use notation like \(\sqrt[3]{a}\) to represent the cube root of \(a\). One essential property of cube roots is that they "undo" cubing. So if \(x = \sqrt[3]{y}\), then \(x^3 = y\).

• Cubing both sides: This technique is used to eliminate the cube root, making it easier to solve the equation.
• Handling negatives: It's important to remember that cube roots can handle negative numbers, unlike square roots.

Understanding cube roots helps us simplify complex equations, especially in cases where the radicals appear on both sides.

Equating Expressions

When we say 'equate expressions,' we refer to setting two expressions equal to each other. This is a core technique in solving equations where similar functions, like cube roots, appear on both sides.

In the original exercise \(\sqrt[3]{-6x-1} = \sqrt[3]{-2x-5}\), we can equate the expressions inside the radicals directly because if two cube roots are equal, the expressions they're derived from must also be equal. This gives us \(-6x - 1 = -2x - 5\).

• Direct equating: This step highlights how simplifying our work becomes possible by removing the radicals through equating.
• Basic algebra: Once the radicals are removed, you simply deal with the resulting linear equation.

Equating expressions is about recognizing opportunities to simplify the mathematics by conceptually reducing the complexity of the problem.

Simplifying Equations

Simplifying equations involves rearranging them to make solutions obvious or easier to find. Once we equate expressions, the next step is simplification, which often involves combining like terms and isolating the variable of interest.

For the equation \(-6x - 1 = -2x - 5\), we rearrange it by moving terms involving \(x\) to one side and constants to the other. By adding \(2x\) to both sides and adjusting constants, we arrive at \(-4x = -4\).

• Combining like terms: This simplifies the equation into a form that's easier to work with.
• Isolating the variable: Solve for \(x\) by dividing by the coefficient of \(x\).

This process results in a straightforward solution, \(x = 1\), showcasing the power of simplifying equations.