Problem 57 - Precalculus Student Solutions Manual 5th

Question

Solve each equation. $$\sqrt[3]{4 x+3}=\sqrt[3]{2 x-1}$$

Step-by-Step Solution

Answer

x = -2

Step 1: Set the cube roots equal to each other

Given that both sides of the equation have cube roots, we can set their radicands equal to each other. Therefore, we have: \ \(4x + 3 = 2x - 1\)

Step 2: Subtract \(2x\) from both sides

To isolate \(x\), subtract \(2x\) from both sides of the equation: \ \(4x - 2x + 3 = 2x - 2x - 1\) \ \Simplifying this, we get: \ \(2x + 3 = -1\)

Step 3: Subtract 3 from both sides

Next, subtract 3 from both sides to further isolate \(x\): \ \(2x + 3 - 3 = -1 - 3\) \ \Simplifying this, we get: \ \(2x = -4\)

Step 4: Divide both sides by 2

Finally, divide both sides by 2 to solve for \(x\): \ \(\frac{2x}{2} = \frac{-4}{2}\) \ \Simplifying this, we get: \ \(x = -2\)

Key Concepts

cube roots

Cube roots are a type of radical that involve raising a number to the power of \(\frac{1}{3}\). Essentially, taking the cube root of a number reverses the process of cubing that number. For example, \(\sqrt[3]{8} = 2\) because \(2^3 = 8\). Cube roots are important in solving equations because they allow us to simplify and compare different expressions.
When dealing with cube roots, it’s important to remember that:

• If \(a = b\), then \(\sqrt[3]{a} = \sqrt[3]{b}\).
• The cube root function is the inverse of the cubing function.

In the given exercise, both sides of the equation have cube roots. This allows us to set the expressions inside the roots equal to each other, simplifying the solving process.

isolating variables

Isolating variables is a fundamental technique in solving equations. The goal is to rearrange the equation so that the unknown variable stands alone on one side. This typically involves a series of operations, such as addition, subtraction, multiplication, or division.
For instance, in the exercise solved:

• First, we set the expressions inside the cube roots equal: \(4x + 3 = 2x - 1\).
• Then, subtract \(2x\) from both sides to begin isolating \(x\): \(4x - 2x + 3 = -1\).
• Next, subtract 3 from both sides: \(2x = -4\).
• Finally, divide by 2 to solve for \(x = -2\).

Each step systematically reduces the complexity of the equation until the variable is isolated.

linear equations

Linear equations are equations of the first degree, meaning they involve only the first power of the variable. The general form is \(ax + b = c\), where \(a, b,\) and \(c\) are constants.
In our example, once the cube roots are removed, the equation is reduced to a linear form \(4x + 3 = 2x - 1\). Solving linear equations typically involves the following steps:

• Combine like terms.
• Isolate the variable on one side of the equation.
• Simplify.

Linear equations are straightforward to solve and serve as a foundation for understanding more complex algebraic equations. They help cement the understanding of balancing both sides of the equation, a critical concept in algebra.