Problem 67 - A Graphical Approach to College Algebra

Question

Solve each rational inequality by hand. Do not use a calculator. $$\frac{1}{x-3} \leq \frac{5}{x-3}$$

Step-by-Step Solution

Answer

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

Step 1: Simplify the Inequality

Begin by simplifying both sides of the inequality. Given equation is \( \frac{1}{x-3} \leq \frac{5}{x-3} \). Both sides have the same denominator, so it reduces to solving \( 1 \leq 5 \), which is always true.

Step 2: Consider the Denominator

The original inequality depends on being defined, so identify where \( x-3 = 0 \), which means \( x eq 3 \). This is a point of discontinuity and should be excluded from the solution.

Step 3: Interpret the Solution Set

Since \( 1 \leq 5 \) is universally true and \( x eq 3 \) due to the denominator, the solution set includes all real numbers except \( x = 3 \). Express this solution set as \( (-\infty, 3) \cup (3, \infty) \).

Key Concepts

Solution Set

A solution set in mathematics refers to the set of all values that satisfy a given equation or inequality. In the context of rational inequalities, it's important to determine which values of the variables make the inequality true. For the inequality \( \frac{1}{x-3} \leq \frac{5}{x-3} \), after simplifying both sides, we find that the inequality always holds true since 1 is indeed less than or equal to 5. However, the solution set is affected by the variable restrictions.

We must exclude \( x = 3 \) from our solution set because it leads to division by zero. Thus, even though \( 1 \leq 5 \) is true for any \( x \), the real number line needs a break at \( x = 3 \). Therefore, the complete solution set is expressed as all real numbers except \( x = 3 \):

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

Familial with writing solution sets in this manner simplifies communicating which values work, while staying clear of those that are undefined.

Discontinuity

In mathematics, a point of discontinuity occurs where a function is not continuous. This can happen when expression within a function makes it undefined or not viable. For rational expressions such as \( \frac{1}{x-3} \), a discontinuity appears whenever the denominator is zero.

For our specific exercise, at \( x=3 \), the denominator \( x-3 \) becomes zero. This makes the function undefined and introduces a discontinuity that we cannot ignore. These points are crucial to identifying when determining the solution set because they are the values where the function does not exist.

When solving inequalities and sketching graphs, highlighting discontinuities ensures you recognize places where the inequality "jumps over" undefined regions. Rational inequalities often involve steps where identifying and excluding these discontinuities is necessary to find the correct solutions.

Inequality Simplification

Simplifying an inequality is just breaking it down to find its simplest form, allowing you to easily determine what values make it true. With rational inequalities, this often involves ensuring both sides have common denominators, like in \( \frac{1}{x-3} \leq \frac{5}{x-3} \).

By having the same denominators, you can compare numerators directly, which simplifies the inequality to a basic comparison: \( 1 \leq 5 \). Simple comparisons often lead to universally true statements which can simplify understanding greatly. Additionally, simplifying helps to quickly see any variable restrictions like discontinuities.

It's important to strike a balance between solving the inequality and keeping track of any restrictions or domain limits these simplifications reveal. Simplification is a helpful tool but must be used carefully to respect all aspects of the inequality.