Problem 71 - College Algebra with Modeling and Visualization

Question

Solve the rational inequality. $$ \frac{1}{x-3} \leq \frac{5}{x-3} $$

Step-by-Step Solution

Answer

The solution is \((3, \infty)\).

Step 1: Understand and rewrite the inequality

To solve the rational inequality \( \frac{1}{x-3} \leq \frac{5}{x-3} \), first subtract \( \frac{5}{x-3} \) from both sides which gives: \( \frac{1}{x-3} - \frac{5}{x-3} \leq 0 \). This simplifies to \( \frac{-4}{x-3} \leq 0 \).

Step 2: Simplify the expression

The expression \( \frac{-4}{x-3} \leq 0 \) implies that the fraction is less than or equal to zero. Since the numerator \(-4\) is negative, the inequality holds when \((x-3)\) is positive or zero. Thus we need \((x-3)\) to be positive.

Step 3: Solve for \(x\)

The condition that \((x-3)\) is positive leads to the inequality \(x-3 > 0\). Solving for \(x\), we get \(x > 3\). However, since the denominator cannot be zero, \(x eq 3\), thus confirming the condition to only include \(x > 3\).

Step 4: Write the solution in interval notation

The solution to the inequality \( \frac{1}{x-3} \leq \frac{5}{x-3} \) is all values of \(x\) that satisfy \( x > 3 \). In interval notation, this solution is \((3, \infty)\).

Key Concepts

Inequality Solving Steps

When solving rational inequalities, the goal is to determine the range of values for the variable that make the inequality true. Here are the steps you can follow:

• Rewrite the Inequality: Start by transforming the inequality into a more manageable form. For example, the original inequality \( \frac{1}{x-3} \leq \frac{5}{x-3} \) can be rewritten by subtracting \( \frac{5}{x-3} \) from both sides to obtain \( \frac{-4}{x-3} \leq 0 \).
• Simplify the Expression: After rewriting, look at the simplified rational expression. Determine its sign by examining the numerator and denominator separately. In this case, the numerator is \(-4\), which is always negative.
• Find the Solution Set: Identify the conditions under which the entire fraction is less than or equal to zero. Since \(-4\) is negative, the fraction \( \frac{-4}{x-3} \leq 0 \) holds true when \( x-3 \) is positive, leading to the inequality \( x > 3 \).
• Consider Domain Restrictions: Ensure that the critical values do not make any part of the fraction undefined. Here, since \( x = 3 \) would make the denominator zero, it must be excluded from the solution set.

Interval Notation

Interval notation provides a concise way to express the set of all solutions to an inequality. It describes intervals of numbers on the real number line.

• Open and Closed Intervals: In interval notation, an open interval \((a, b)\) includes all numbers greater than \( a \) and less than \( b \), but not \( a \) and \( b \) themselves. A closed interval \([a, b]\) includes all numbers as well as the endpoints \( a \) and \( b \).
• Infinite Intervals: When intervals extend infinitely, we use \( \infty \) and \( -\infty \) to represent them. The symbol \( \infty \) is always accompanied by a parenthesis, indicating that it is not an actual number.
• Specific to our case: The solution \( x > 3 \) is expressed as an interval \((3, \infty)\), which includes all numbers greater than 3 but excludes 3 itself.

Rational Expressions

Rational expressions are quotients of polynomials, similar to fractions, with a numerator and denominator.

• Components: A rational expression like \( \frac{1}{x-3} \) has a numerator \( 1 \) and a denominator \( x-3 \). These components determine its value for different \( x \).
• Finding the Domain: It is crucial to remember that the denominator cannot be zero, as this would make the expression undefined. To find the domain of a rational expression, exclude values of \( x \) that make the denominator zero.
• Analyzing Signs: When solving inequalities involving rational expressions, check the sign of the expression over different intervals determined by the denominator's critical points. These are points where the denominator changes from positive to negative or vice versa.