Problem 44 - Algebra and Trigonometry with Analytic Geometry

Question

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

Step-by-Step Solution

Answer

The solution is \((-\infty, 2)\).

Step 1: Identify the Condition for Negative Fraction

The fraction \( \frac{-3}{2-x} \) is negative when the numerator and the denominator have opposite signs. The numerator, \(-3\), is already negative, so for the fraction to be negative overall, the denominator \(2-x\) must be positive.

Step 2: Solve the Inequality for the Denominator

Set up the inequality for the denominator to be positive by solving \( 2 - x > 0 \). This results in \( 2 > x \) or \( x < 2 \).

Step 3: Express the Solution as an Interval

The solution \( x < 2 \) can be expressed in interval notation as \((-\infty, 2)\). This indicates that any value of \(x\) less than 2 will satisfy the inequality.

Key Concepts

Numerators

In any fraction, the numerator is the number on the top. It's important to understand its role, especially in terms of solving inequalities. A negative numerator inherently affects the fraction's value. In our example, the numerator is -3. Because it's negative, it determines the overall sign of the fraction together with the denominator's sign. When you are solving an inequality with a fraction, pay close attention to whether the numerator is positive or negative, as this will dictate the conditions under which the fraction itself remains negative or positive. This can often give you a valuable clue as to the specific inequality conditions needed for solving.

Denominators

The denominator is the bottom part of a fraction. It's essential to know how its value influences the fraction. If the denominator is positive, it means that the fraction's value is aligned with the numerator's sign. If it's negative, it causes the fraction to flip its sign.
In our exercise, the denominator is given by the expression \(2-x\). To ensure the fraction \(\frac{-3}{2-x}<0\) holds true, the denominator \(2-x\) must be positive, due to the negative numerator. This condition allows us to derive that \(2-x > 0\), which simplifies to \(x < 2\).
Remember, the role of the denominator in an inequality is crucial, as it essentially governs the conditions under which the inequality will either be satisfied or not.

Interval Notation

Interval notation is a simplified way of writing the set of solutions or intervals where a condition or inequality is satisfied. It uses parentheses and square brackets to define ranges.
For example, the inequality \(x < 2\) can be expressed as \((-\infty, 2)\) using interval notation. The round parenthesis near \(2\) signifies that \(2\) is not included in the solution set, which aligns with the fact that \(x\) is strictly less than \(2\).
Interval notation is a very concise and intuitive way to represent solutions, especially when dealing with continuous ranges of values, as it makes it easy to understand which numbers are included in the solution.

Inequality Conditions

Inequality conditions are the specific requirements that must be met for an inequality to hold true. In our example, for \(\frac{-3}{2-x} < 0\), we have to ensure that the numerator and denominator have opposite signs for the fraction to be negative.
This was achieved by understanding that a negative numerator requires a positive denominator. Hence, we solved \(2-x > 0\) to find that \(x < 2\).
Understanding inequality conditions involves recognizing the interplay between numerators and denominators, especially when dealing with fractions. Correctly identifying these conditions helps in determining the appropriate solution set for the inequality.