Problem 45 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

The solution is \( x = -15 \).

Step 1: Understand the Problem

We need to solve the equation \( \sqrt[3]{2x + 3} = -3 \). This means finding the value of \( x \) that makes this equation true.

Step 2: Remove the Cube Root

To eliminate the cube root, we will cube both sides of the equation. Cubing the cube root cancels it out, giving: \[ (\sqrt[3]{2x + 3})^3 = (-3)^3 \]This simplifies to: \[ 2x + 3 = -27 \]

Step 3: Solve for x

Now, solve the linear equation for \( x \). First, subtract 3 from both sides: \[ 2x = -27 - 3 \]\[ 2x = -30 \]Next, divide both sides by 2: \[ x = \frac{-30}{2} \]\[ x = -15 \]

Step 4: Check the Solution

Substitute \( x = -15 \) back into the original equation to verify: \[ \sqrt[3]{2(-15) + 3} = \sqrt[3]{-30 + 3} = \sqrt[3]{-27} \]Since \( \sqrt[3]{-27} = -3 \), which matches our original equation, \( x = -15 \) is the correct solution.

Key Concepts

Cube Roots

Cube roots are the inverse operation of cubing a number. When you take a cube root of a number, you're essentially finding the value that, when multiplied by itself three times, gives you the original number. In our particular problem, we need to eliminate the cube root to solve for \( x \).

Here’s how you can think about cube roots:

• Cube root of 8 is 2, because \( 2 \times 2 \times 2 = 8 \).
• Cube root of -27 is -3, because \( -3 \times -3 \times -3 = -27 \).

Cube roots can produce both positive and negative results. In linear equations like \( \sqrt[3]{2x + 3} = -3 \), identifying the cube root helps simplify the equation by cancelling both sides. This allows us to isolate and solve for \( x \).

Remember, cubing both sides of an equation safely removes the cube root, allowing you to freely work with the resulting expression.

Linear Equations

Linear equations are the simplest type of equations because they represent straight lines when graphed. They have the general form \( ax + b = c \). Solving linear equations involves basic algebraic operations that aim to isolate the variable, making it simpler to figure out its value.

For example, in our case, once we got rid of the cube root, the equation \( 2x + 3 = -27 \) emerged. Solving this linear equation consisted of basic steps:

• First, isolate the term with \( x \) by subtracting 3 from each side.
• Then, divide the entire equation by 2 to solve for \( x \).

The operation does not change because the properties of equalities ensure that whatever you do to one side, you should do to the other. By following these steps and keeping balance, solving for \( x \) becomes straightforward.

Equation Verification

Once you obtain a potential solution for an equation, verifying it is crucial to ensure there are no mistakes in your calculations. In the problem \( \sqrt[3]{2x + 3} = -3 \), after simplifying and solving the equation, we need to confirm that \( x = -15 \) is correct.

Verification involves substituting the found value of \( x \) back into the original equation:

• Substitute \( x = -15 \) into \( 2x + 3 \): \[ 2(-15) +3 = -30 + 3 = -27 \]
• Check if the cube root of \(-27\) is \(-3\), which it is.

This confirms that our solution is accurate because it satisfies the initial equation. Verifying your solution not only checks for errors but also increases your confidence in your answer. It’s an important step in solving equations that should never be overlooked.