Problem 9 - Algebra and Trigonometry

Question

Determine whether the given value is a solution of the equation. \(2 x^{1 / 3}-3=1\) (a) \(x=-1 \quad\) (b) \(x=8\)

Step-by-Step Solution

Answer

(a) No, (b) Yes.

Step 1: Substitute x = -1 into the equation

First, substitute the value of \(x = -1\) into the original equation \(2x^{1/3} - 3 = 1\). We get: \[2(-1)^{1/3} - 3 = 1\].

Step 2: Simplify the expression for x = -1

Calculate \((-1)^{1/3}\), which is the cube root of -1. The cube root of -1 is -1. Now, substitute back: \[2(-1) - 3 = 1\] which simplifies to \[-2 - 3 = -5\]. Since \(-5 eq 1\), \(x = -1\) is not a solution.

Step 3: Substitute x = 8 into the equation

Now, substitute \(x = 8\) into the original equation \(2x^{1/3} - 3 = 1\). We get: \[2(8)^{1/3} - 3 = 1\].

Step 4: Simplify the expression for x = 8

Calculate \((8)^{1/3}\), which is the cube root of 8. The cube root of 8 is 2. Now substitute back: \[2 \times 2 - 3 = 1\] which simplifies to \[4 - 3 = 1\]. Since \(1 = 1\), \(x = 8\) is a solution.

Key Concepts

Equation Solving

Equation solving is a fundamental skill in algebra that involves finding values for the variables that make an equation true. An equation is like a balance scale, where both sides must have the same value. Solving equations means adjusting the values of variables to maintain that balance.

When solving equations, keep these points in mind:

• Identify the equation to solve.
• Simplify the equation if necessary (combine like terms or eliminate fractions).
• Isolate the variable by rearranging the equation.
• Verify the solution by substituting the variable back into the original equation.

In our example, we are determining if given values (like -1 and 8) satisfy the equation. By substituting and simplifying, we check if the left-hand side of the equation equals the right-hand side for each value.

Cube Roots

Understanding cube roots is crucial when dealing with equations involving powers of three. A cube root of a number is a value that, when multiplied by itself three times, gives the original number. For example, the cube root of 8 is 2 because 2 \times 2 \times 2 = 8.

Here are key points about cube roots:

• The cube root of a number \(x\) is denoted as \(x^{1/3}\).
• Cube roots can be negative; for instance, \((-1)^{1/3} = -1\).
• They are applied when solving equations where the variable is raised to the power of three, as seen in the equation \(2x^{1/3} - 3 = 1\).

In our problem, calculating the cube root is a step in determining the solution for each potential value \(x\), aiding in verifying the equation's truth.

Substitution Method

The substitution method is a simple technique used to evaluate whether a specific value is a solution for an equation. It involves replacing the variable with the given number in the equation and checking the result.

Steps to apply the substitution method include:

• Take the value you need to check and substitute it directly into the equation in place of the variable.
• Simplify the equation results to see if both sides of the equation become equal.
• If they are equal, the value is a solution; if not, the value is not a solution.

In the exercise, substituting \(x = -1\) didn't satisfy the equation, while \(x = 8\) did, demonstrating how this method confirms which values solve the equation.