Problem 36 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-40: Solve the inequality, and express the solutions in terms of intervals whenever possible. $$ \frac{3}{5 x+1} \geq \frac{1}{x-3} $$

Step-by-Step Solution

Answer

The solution interval is \(( -\infty, -5 ]\).

Step 1: Understand the Inequality

The inequality to solve is \( \frac{3}{5x+1} \geq \frac{1}{x-3} \). This means we are comparing two rational expressions.

Step 2: Find the common denominator

To solve the inequality \( \frac{3}{5x+1} \geq \frac{1}{x-3} \), find a common denominator to combine them. The common denominator is \((5x+1)(x-3)\).

Step 3: Combine the expressions

Multiply each side by the common denominator \((5x+1)(x-3)\), resulting in: \(3(x-3) - (5x+1) \geq 0\). Simplify to get a single inequality.

Step 4: Simplify the expression

Simplifying the expression, we get \(3x - 9 - 5x - 1 \geq 0\), which simplifies to \(-2x - 10 \geq 0\).

Step 5: Solve for x

To solve \(-2x - 10 \geq 0\), add 10 to both sides to get \(-2x \geq 10\). Divide by \(-2\) (and remember to flip the inequality), resulting in \(x \leq -5\).

Step 6: Consider Invalid Points

Note that \(x = -\frac{1}{5}\) and \(x = 3\) are invalid as they make the denominators zero. So, these points must be excluded.

Step 7: Determine Valid Intervals

The solution from the inequality is \(x \leq -5\). However, you also have exclusions at \(x = -\frac{1}{5}\) and \(x = 3\) from the invalidation points, but \(x \leq -5\) doesn't include them.

Step 8: Express solution in interval notation

The valid solution set considering invalid points and the inequality is \(( -\infty, -5 ]\).

Key Concepts

Rational Expressions

Rational expressions are fractions where the numerator and/or the denominator are polynomials. In the inequality \( \frac{3}{5x+1} \geq \frac{1}{x-3} \), each side consists of a rational expression. The important aspect here is the behavior of these expressions across different values of \( x \).
We must pay special attention to the values that make the denominator zero, as they are undefined and need to be excluded from the solution set. With \( 5x+1 \) and \( x-3 \) as denominators, the critical points are \( x = -\frac{1}{5} \) and \( x = 3 \).
Understanding how to handle these undefined points is key to correctly solving the inequality without errors.

Interval Notation

Interval notation is a concise way to describe a range of numbers, typically expressed as intervals on the number line. For example, the solution to our inequality is \(( -\infty, -5 ]\).
In interval notation, round brackets \( () \) are used to indicate that an endpoint is not included, while square brackets \( [] \) show that an endpoint is included. So, \(( -\infty, -5 ]\) means that all numbers less than or equal to \(-5\) are included, but \(-5\) is the endpoint.
This shorthand is incredibly useful when expressing solutions to inequalities because it provides a clear range of solutions that are easy to understand and visually represent on a number line.

Common Denominator

When dealing with rational expressions and inequalities, finding a common denominator is crucial. It allows us to combine the expressions effectively. For the inequality \( \frac{3}{5x+1} \geq \frac{1}{x-3} \), the common denominator is \((5x+1)(x-3)\).
By multiplying each term by this denominator, we eliminate the fractions and work with a polynomial inequality instead. This transformation simplifies our work immensely, reducing potential algebraic errors.
Always ensure that you multiply out expressions carefully and double-check for signs or terms that can change the inequality when you start simplifying after clearing the denominators.

Solving Inequalities

Solving inequalities involves finding the set of all possible values that satisfy the inequality condition. For \( -2x - 10 \geq 0 \), solving it properly is key.
Start by isolating \( x \) through basic algebraic operations like adding and dividing. When dividing by a negative number, remember to flip the inequality sign. This step is crucial, as it can change the direction of the inequality, leading to a different set of solutions.

• First, add \( 10 \) to both sides: \( -2x \geq 10 \).
• Second, divide by \(-2\): \( x \leq -5 \).

Finally, consider any restrictions from the domain of the original rational expressions, excluding any points where the original expressions are undefined. In our example, \(-\frac{1}{5}\) and \(3\) must be excluded, even if the inequality solution does not seem to include them directly.