Problem 29 - Precalculus

Question

Solve the equation or inequality. $$2(x-2)^{-\frac{1}{3}}-\frac{2}{3} x(x-2)^{-\frac{4}{3}} \leq 0$$

Step-by-Step Solution

Answer

Solution: \(2 < x \leq 3\).

Step 1: Identifying the common factor

Identify and take out the common factor from both terms. In this case, both terms have \((x-2)^{-\frac{4}{3}}\) as a common factor. Factor the inequality to get:\[(x-2)^{-\frac{4}{3}} \left[ 2(x-2) - \frac{2}{3}x \right] \leq 0.\]

Step 2: Simplifying the expression

Simplify the expression inside the brackets. The expression is:\[ 2(x-2) - \frac{2}{3}x = 2x -4 - \frac{2}{3}x = \frac{6}{3}x - \frac{2}{3}x - 4 = \frac{4}{3}x - 4.\]So the inequality becomes:\[(x-2)^{-\frac{4}{3}} \left( \frac{4}{3}x - 4 \right) \leq 0.\]

Step 3: Finding zero points and critical points

Identify the zero points and points where the expression is undefined. The expression \((x-2)^{-\frac{4}{3}}\) is undefined at \(x = 2\). The expression \(\frac{4}{3}x - 4 = 0\) gives \(x = 3\). Thus, the critical points are \(x = 2\) and \(x = 3\).

Step 4: Testing intervals

Test the intervals determined by critical points: \(( -\infty, 2)\), \((2, 3)\), and \((3, \infty)\). Choose test points such as \(x = 0\), \(x = 2.5\), and \(x = 4\): - For \(x = 0\): \( (0-2)^{-\frac{4}{3}} \left(\frac{4}{3}(0) - 4\right) > 0\).- For \(x = 2.5\): \( (2.5-2)^{-\frac{4}{3}} \left(\frac{4}{3}(2.5) - 4\right) < 0\).- For \(x = 4\): \( (4-2)^{-\frac{4}{3}} \left(\frac{4}{3}(4) - 4\right) > 0\).

Step 5: Interpreting testing results

For the inequality \((x-2)^{-\frac{4}{3}} \left( \frac{4}{3}x - 4 \right) \leq 0\), the expression is non-positive in the interval \((2, 3]\). Therefore, the solution to the inequality is \(2 < x \leq 3\).

Key Concepts

Common Factor

The process of finding a common factor in an expression helps simplify complex terms, making them easier to handle. It involves identifying a term that appears in each part of the expression. In the given inequality, both terms contain \((x-2)^{-\frac{4}{3}}\) as a common factor.
This element is factored out, simplifying the inequality altogether.

By factoring, we can express the inequality as \((x-2)^{-\frac{4}{3}}[2(x-2) - \frac{2}{3}x] \leq 0\).
Such simplification is crucial because it prepares the expression for further analysis.
Factoring out the common term reduces potential complexity and allows the identification of critical points more easily.

Critical Points

Critical points in the inequality include points where the expression equals zero or becomes undefined.
These points are essential as they dictate the boundaries of the solution interval.

To determine critical points, examine both terms of the simplified inequality:

• The expression \((x-2)^{-\frac{4}{3}}\) becomes undefined at \(x = 2\).
• The expression \(\frac{4}{3}x - 4 = 0\) results in a critical point at \(x = 3\).

Critical points \(x = 2\) and \(x = 3\) break the number line into testable intervals.
These points help us focus on specific intervals that determine the inequality's solution.

Interval Testing

Interval testing is a strategy to determine where an inequality holds true.
After finding critical points, these points define intervals that need testing.

In this problem, critical points \(x = 2\) and \(x = 3\) create three intervals:

• \((-\infty, 2)\)
• \((2, 3)\)
• \((3, \infty)\)

To test each interval, choose any \(x\) value within the interval and substitute it into the inequality.
For instance:

• \(x = 0\) in \((-\infty, 2)\) yields a positive result.
• \(x = 2.5\) within \((2, 3)\) results in a negative value.
• \(x = 4\) in \((3, \infty)\) gives a positive outcome.

This analysis reveals that the given inequality holds only in the interval \((2, 3]\).
Thus, interval testing offers a clear picture of truly satisfied regions.

Undefined Expression

An undefined expression in mathematics occurs when substituting a certain value results in an impossibility, like division by zero.
These undefined points are crucial for analyzing expressions and inequalities.

In our inequality, the term \((x-2)^{-\frac{4}{3}}\) becomes undefined if \(x = 2\) because it results in division by zero.
This makes \(x = 2\) a critical point.

Understanding when an expression is undefined prevents confusion when solving inequalities.
By recognizing these points, you avoid making incorrect assumptions about the expression's value.
Moreover, acknowledging undefined expressions also assists in correctly identifying intervals where the inequality is valid.