Solving Inequalities Expressing Solutions In Interval And Set Notation

Welcome to an in-depth exploration of solving inequalities, where we'll dissect three distinct problems, each demanding a comprehensive solution encompassing interval notation, set notation, and graphical representation. Inequalities, unlike equations, deal with relationships that are not strictly equal, opening up a range of solutions rather than a single point. This exploration is crucial in various fields, from economics modeling supply and demand curves to physics analyzing motion within certain constraints. Mastering these techniques not only enhances your mathematical prowess but also equips you with a powerful tool for real-world problem-solving.

1. Solving ${ \frac{x+3}{3x-6} > 0 }$

Unveiling the Solution: A Step-by-Step Approach

To solve the inequality ${ \frac{x+3}{3x-6} > 0 }$, our primary goal is to identify the range of ${ x }$ values that satisfy this condition. This involves a methodical approach, focusing on critical points and interval testing. Critical points are the values of ${ x }$ that make either the numerator or the denominator of the fraction equal to zero. These points are crucial because they are where the expression can change its sign. This first step will give us a framework for understanding where the expression is positive or negative. In this case, the critical points will define the intervals we need to test. Remember, we're looking for where the entire fraction is greater than zero, meaning it's positive. The critical points will act as dividers on the number line, helping us pinpoint these positive intervals. Each critical point represents a potential boundary where the inequality's solution changes, making them indispensable for accurate solving.

First, we need to find the critical points. We set the numerator and the denominator equal to zero and solve for ${ x }$:

• Numerator: ${ x + 3 = 0 }$ => ${ x = -3 }$
• Denominator: ${ 3x - 6 = 0 }$ => ${ x = 2 }$

These critical points, ${ x = -3 }$ and ${ x = 2 }$, divide the number line into three intervals: ${ (-\infty, -3) }$, ${ (-3, 2) }$, and ${ (2, \infty) }$. We will now test a value from each interval in the original inequality to determine where the inequality holds true. Interval testing is the cornerstone of solving inequalities because it allows us to systematically check each region created by the critical points. By substituting a test value from within each interval into the original inequality, we can determine whether that entire interval satisfies the condition. This method efficiently identifies which ranges of ${ x }$ values make the inequality true or false, providing a clear picture of the solution set. It's a fundamental technique that converts a complex inequality problem into a series of manageable checks. This systematic approach ensures we don't miss any part of the solution and accurately capture the behavior of the inequality across the number line.

• Interval ${ (-\infty, -3) }$: Let's test ${ x = -4 }$: ${ \frac{-4 + 3}{3(-4) - 6} = \frac{-1}{-18} = \frac{1}{18} > 0 }$ (True)
• Interval ${ (-3, 2) }$: Let's test ${ x = 0 }$: ${ \frac{0 + 3}{3(0) - 6} = \frac{3}{-6} = -\frac{1}{2} > 0 }$ (False)
• Interval ${ (2, \infty) }$: Let's test ${ x = 3 }$: ${ \frac{3 + 3}{3(3) - 6} = \frac{6}{3} = 2 > 0 }$ (True)

Therefore, the solution includes the intervals ${ (-\infty, -3) }$ and ${ (2, \infty) }$.

Expressing the Solution: Interval Notation, Set Notation, and Graph

Now, let's represent our solution in three different forms, each offering a unique perspective on the answer. Interval notation is a concise way to describe a set of numbers that lie within a specific range. It uses parentheses and brackets to indicate whether the endpoints are included or excluded, providing a clear and efficient way to represent solutions to inequalities. Set notation, on the other hand, uses curly braces and mathematical symbols to define the solution set, offering a more formal and precise representation. It's particularly useful for describing solutions that involve unions and intersections of intervals. Lastly, a graphical representation visually displays the solution set on a number line, making it easy to understand the range of values that satisfy the inequality. This visual approach is invaluable for gaining an intuitive understanding of the solution and for communicating it effectively.

• Interval Notation: ${ (-\infty, -3) \cup (2, \infty) }$
• Set Notation: ${ \{ x \mid x < -3 \text{ or } x > 2 \} }$
• Graph: A number line with open circles at -3 and 2, and the regions to the left of -3 and to the right of 2 shaded.

2. Tackling ${ \frac{x-4}{x+5} \le 0 }$

Solving the Inequality: Critical Points and Interval Testing

Moving on to the inequality ${ \frac{x-4}{x+5} \le 0 }$, we employ a similar strategy, but with a crucial difference: we're looking for values where the expression is less than or equal to zero. This means we need to carefully consider whether the critical points themselves are included in the solution. The inclusion or exclusion of critical points is a pivotal aspect of solving inequalities. When an inequality includes