Problem 41 - A Graphical Approach to College Algebra

Question

Use an analytic method to solve each equation in part (a). Support the solution with a graph. Then use the graph to solve the inequalities in parts (b) and (c). (a) \(\sqrt[3]{x^{2}-2 x}=\sqrt[3]{x}\) (b) \(\sqrt[3]{x^{2}-2 x}>\sqrt[3]{x}\) (c) \(\sqrt[3]{x^{2}-2 x}<\sqrt[3]{x}\)

Step-by-Step Solution

Answer

(a) x = 0, x = 3; (b) (0, 3); (c) (-∞, 0).

Step 1: Simplify the Equation

First, we want to solve the equation \(\sqrt[3]{x^2 - 2x} = \sqrt[3]{x}\). We can cube both sides to eliminate the cube roots: \[(x^2 - 2x) = x\] This simplifies the equation to \(x^2 - 2x = x\).

Step 2: Rearrange and Solve for x

Rearrange the equation to bring all terms to one side: \[x^2 - 2x - x = 0\] This simplifies to: \[x^2 - 3x = 0\] Factor out an \(x\): \[x(x - 3) = 0\] Set each factor to zero: 1. \(x = 0\)2. \(x - 3 = 0\Rightarrow x = 3\) Therefore, the solutions to the equation are \(x = 0\) and \(x = 3\).

Step 3: Analyze the Graph

The graph of \(y = \sqrt[3]{x^2 - 2x}\) and \(y = \sqrt[3]{x}\) will intersect at the solutions we found, \(x = 0\) and \(x = 3\). By plotting these functions, you'll notice they intersect at these points, confirming the solutions.

Step 4: Solve the Inequality (b)

For \(\sqrt[3]{x^2 - 2x} > \sqrt[3]{x}\), check intervals based on critical points found: \(-\infty, 0, 3, +\infty\). - For \(x = -1\) (in \((-\infty, 0)\)), \(-\frac{1}{3} > 0\) is false. - For \(x = 1\) (in \((0, 3)\)), \(-\frac{2}{3} < \frac{1}{3}\) is true.- For \(x = 4\) (in \((3, \infty)\)), \(\frac{8}{3} > \frac{4}{3}\) is false.The solution is the interval \((0, 3)\).

Step 5: Solve the Inequality (c)

For \(\sqrt[3]{x^2 - 2x} < \sqrt[3]{x}\), examine the same intervals:- In \((-\infty, 0)\), \(-\frac{1}{3} < 0\), which is true.- In \((0, 3)\), \(-\frac{2}{3} > \frac{1}{3}\) is false.- In \((3, +\infty)\), \(\frac{8}{3} < \frac{4}{3}\) is false.Thus, the solution is \((-\infty, 0)\). The graph supports these by showing where one function is above or below the other.

Key Concepts

Inequalities

Understanding inequalities is crucial when comparing two expressions to see which one is larger or smaller. In mathematics, inequalities often require you to solve for which values of the variable satisfy the given condition.
Given the inequality

• \( \sqrt[3]{x^2 - 2x} > \sqrt[3]{x} \)
• this implies that you need to find the intervals where the cube root expression on the left is greater than the cube root expression on the right.

When solving such inequalities, dividing the number line into intervals around known solutions helps. You test points within these intervals to determine if they satisfy the inequality. It's like splitting the problem into manageable parts because the behavior of expressions can change dramatically across different intervals.
Always remember to express your final answers as intervals, showing where the inequality holds true.

Cube Roots

Cube roots are the opposite of cubing a number and they are not limited by sign, unlike square roots. If you cube 3, you get 27. Thus, the cube root of 27 is 3. In algebra, cube roots help simplify equations, just like in the given problem with

• \( \sqrt[3]{x^2 - 2x} = \sqrt[3]{x} \).

To solve this, cubing both sides is a powerful technique because it removes the cube roots, allowing you to work with the polynomials directly. Once simplified, the problem becomes a regular quadratic equation,

• \( x^2 - 3x = 0 \).

Factoring this equation gives solutions easily. Cube roots are useful in these problems because they provide symmetry and a specific domain where solutions lie. Unlike square roots, cube roots always give real numbers, which simplifies considerations about the kind of numbers you are working with.

Graphical Analysis

Graphical analysis involves using graphs to understand relationships between variables visually. When graphing equations or inequalities like

• \( y = \sqrt[3]{x^2 - 2x} \)
• and \( y = \sqrt[3]{x} \),

you can see where they intersect, indicating where the equations are equal.
The graphs show the solution to the equation. The points of intersection are the values of \( x = 0 \) and \( x = 3 \).
For inequalities, the region where one graph is above or below another determines where the inequality holds. Graphs help to double-check solutions found algebraically and give insight into the behavior of functions over different intervals.
Using both algebraic and graphical methods provides a complete understanding of the problem, ensuring accuracy and reinforcing the concept.