Problem 49 - Algebra for College Students

Question

Solve each inequality. $$\frac{x-1}{x-5} \leq 2$$

Step-by-Step Solution

Answer

The solution is \(( -\infty, 5) \cup [9, \infty)\)."

Step 1: Move Terms to One Side

First, we want to move all terms involving the variable on one side of the inequality. Begin by subtracting 2 from both sides to obtain: \( \frac{x-1}{x-5} - 2 \leq 0 \).

Step 2: Simplify the Expression

Find a common denominator for the fraction on the left side. The common denominator is \(x-5\). Rewrite the inequality as: \( \frac{x-1 - 2(x-5)}{x-5} \leq 0 \).

Step 3: Distribute and Combine Like Terms

Distribute the \(2\) across \( (x-5) \) in the numerator to get: \( \frac{x-1 - 2x + 10}{x-5} \leq 0 \), which simplifies to: \( \frac{-x + 9}{x-5} \leq 0 \).

Step 4: Identify Critical Points

The inequality \( \frac{-x + 9}{x-5} \leq 0 \) is undefined at \(x = 5\). The numerator is zero at \(x = 9\). The critical points are \(x = 5\) and \(x = 9\).

Step 5: Test Intervals Between Critical Points

Test intervals determined by the critical points \(x = 5\) and \(x = 9\):1. For \((-\infty, 5)\), choose \(x = 0\): \(\frac{-0 + 9}{0-5} = -\frac{9}{5} < 0\).2. For \((5, 9)\), choose \(x = 6\): \(\frac{-6 + 9}{6-5} = 3 > 0\).3. For \((9, \infty)\), choose \(x = 10\): \(\frac{-10 + 9}{10-5} = -\frac{1}{5} < 0\).

Step 6: Consider Equality and Exclusion of Undefined Points

Include \(x = 9\) since the inequality is \(\leq\) and \(-x+9 = 0\) at \(x = 9\). Exclude \(x = 5\) because the inequality is undefined here. The solution is \(( -\infty, 5) \cup [9, \infty)\).

Key Concepts

Critical Points

To solve inequalities like \[ \frac{x-1}{x-5} \leq 2 \]understanding critical points is essential. Critical points are values of \(x\) where the function changes its behavior. Specifically, these include:

• Values that make the numerator zero.
• Values that make the denominator zero, causing the expression to be undefined.

In this problem, the inequality becomes undefined at \(x = 5\) because it makes the denominator zero. Additionally, the numerator reaches zero at \(x = 9\) since it nullifies the expression.

Identifying these critical points helps determine the intervals on the number line where we should test the inequality. They divide the domain into separate regions. Understanding their role is crucial since they dictate where the expression is zero or undefined, guiding us to accurately interpret the inequality.

Test Intervals

Once we've identified critical points, the next step involves testing intervals. These intervals are segments of the number line that the critical points divide the domain into. Here, from the example, we have intervals \((-\infty, 5)\), \((5, 9)\), and \((9, \infty)\).

In each interval, pick a point and substitute it back into \[ \frac{-x + 9}{x-5} \]This action allows us to check if the inequality holds in each section.

• For \((-\infty, 5)\), choosing \(x = 0\) gives a negative result.
• For \((5, 9)\), choosing \(x = 6\) results in a positive value.
• For \((9, \infty)\), choosing \(x = 10\) also results in a negative value.

The testing process confirms which intervals make the inequality true. Computational checks allow us to confidently state where the expression is non-positive, guiding to the solution set.

Common Denominator

Utilizing a common denominator simplifies the process of solving inequalities, particularly those involving fractions. In this case, to handle \[ \frac{x-1}{x-5} - 2 \leq 0 \]we align the fractions using \(x-5\) as the common denominator.

This leads to:\[ \frac{x-1 - 2(x-5)}{x-5} \leq 0 \]By adopting a common denominator, we smoothly manage subtraction across fractions and unify them into a single expression.

This approach is pivotal because it reframes the problem in terms of a single rational expression, making it easier to work with. Finding a common denominator efficiently consolidates multiple terms into one manageable component, rendering the inequality ready for further simplification.

Simplifying Expressions

After establishing a common denominator, simplifying expressions ensures we are equipped to solve the inequality properly. Here, distribute and combine like terms in the numerator as follows:\[ \frac{x-1 - 2(x-5)}{x-5} \]Unfold this to get:\[ \frac{x-1 - 2x + 10}{x-5} \]Simplify further, yielding:\[ \frac{-x + 9}{x-5} \leq 0 \] Simplifying reveals a clearer form of the expression.

The act of distributing and combining like terms efficiently reduces the complexity of the expression. This step is crucial because it transforms a cumbersome mathematical statement into an approachable form that is easier to interpret and solve. Ease in simplifying also aids in correctly identifying critical points and interpreting test intervals in pursuit of the correct solution.