Problem 43 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 21-70: Solve the inequality, and express the solutions in terms of intervals whenever possible. $$ \frac{-2}{4-3 x}>0 $$

Step-by-Step Solution

Answer

The solution in interval notation is \( \left( \frac{4}{3}, \infty \right) \).

Step 1: Analyze the Inequality

You need to solve the inequality \( \frac{-2}{4-3x} > 0 \). This inequality asks when the fraction is greater than zero. Remember that a fraction is greater than zero when the numerator and denominator have opposite signs.

Step 2: Determine the Sign of the Numerator

The numerator of the fraction is \(-2\), which is always negative. Therefore, for the fraction to be positive, the denominator \(4 - 3x\) must be negative.

Step 3: Solve for When the Denominator is Negative

Set up the inequality for the denominator: \[ 4 - 3x < 0 \]Solve for \(x\):1. Subtract 4 from both sides: \[ -3x < -4 \]2. Divide both sides by -3, remembering to reverse the inequality sign since you are dividing by a negative: \[ x > \frac{4}{3} \]

Step 4: Express the Solution as an Interval

The solution \( x > \frac{4}{3} \) means that any \(x\) value greater than \(\frac{4}{3}\) will satisfy the inequality. This can be written in interval notation as \( \left( \frac{4}{3}, \infty \right) \).

Key Concepts

Solving Inequalities

Solving inequalities is a fundamental aspect of algebra that involves finding the values of a variable that make the inequality true. Unlike equations, which state that two expressions are equal, inequalities express a range of possible values where one side is greater or less than the other. The symbols used include:

• \( < \) for less than
• \( > \) for greater than
• \( \leq \) for less than or equal to
• \( \geq \) for greater than or equal to

Let's take the inequality from our exercise, \( \frac{-2}{4-3x} > 0 \). We want to determine the set of all \( x \) values that make this statement true. To solve this, we examine both the numerator and denominator. As a basic rule, for the fraction to be positive, the numerator and denominator must have opposite signs. Since \(-2\) is always negative, \(4-3x\) has to be negative for the fraction to be positive.

In performing arithmetic operations on inequalities, remember to flip the inequality sign when multiplying or dividing by a negative number. This foundational step ensures all solutions are correctly oriented, ultimately enabling us to identify the range of \( x \) values satisfying the inequality.

Interval Notation

Interval notation provides a concise way to describe a set of numbers within a particular range. It's utilized extensively in mathematics to represent solutions to inequalities and can be visualized on a number line. In interval notation:

• Parentheses \( () \) indicate that an endpoint is not included (open interval).
• Brackets \( [] \) indicate that an endpoint is included (closed interval).

For the given inequality \( x > \frac{4}{3} \), the endpoint \( \frac{4}{3} \) is not included because it does not satisfy the inequality precisely (since at \( \frac{4}{3} \), the denominator equals zero, which is undefined). Therefore, we write the solution in interval notation as \( \left( \frac{4}{3}, \infty \right) \). The use of a parenthesis here rather than a bracket makes it clear that \( x \) must be strictly greater than \( \frac{4}{3} \).

Understanding interval notation well is crucial as it effectively communicates complex ideas in a simplified format. This notation is not only used for basic inequalities but is also essential in higher-level mathematics.

Rational Inequalities

Rational inequalities involve expressions that contain rational functions—where both the numerator and denominator are polynomials. Solving them involves special considerations compared to linear inequalities due to the presence of variables in the denominator. For instance, \( \frac{-2}{4-3x} > 0 \) features a rational expression.

To solve rational inequalities, you need to:

• Analyze the sign of the numerator and denominator separately.
• Determine where the rational expression is undefined, typically where the denominator equals zero.
• Write and solve one or more inequalities to find critical points.
• Consider intervals between these critical points to check where the inequality holds true.

For our specific inequality, since \(-2\) is negative across all values, we focus on \(4-3x\). By setting \(4-3x \leq 0\), we find critical values by solving: \( x > \frac{4}{3} \). Evaluating these conditions tells us the inequality holds true for \( x \) values greater than \( \frac{4}{3} \), leading us to conclude with confidence that \( x \) in \( \left( \frac{4}{3}, \infty \right) \) is the solution.