Problem 59 - A Graphical Approach to Precalculus with Limits

Question

Use analytic or graphical methods to solve the inequality. $$\sqrt{-x}<0$$

Step-by-Step Solution

Answer

No solution exists in the real numbers.

Step 1: Understanding the Inequality

First, let's recall that the square root of any non-negative number is always non-negative. Therefore, \( \sqrt{-x} \) will either be non-negative or undefined when \(-x\) is negative (meaning \(x\) is positive). To explore whether \( \sqrt{-x} < 0 \) is possible, we need to first identify the domain.

Step 2: Identifying the Domain

For \( \sqrt{-x} \) to be defined, \(-x\) must be greater than or equal to 0, so \(x \leq 0\). This restriction comes from the fact that the square root of a negative number is not defined in the set of real numbers. Therefore, we can restrict our analysis to \(x \leq 0\).

Step 3: Checking the Inequality

Knowing \( \sqrt{-x} \geq 0 \) based on its definition for \(-x \geq 0\), we now find that there's no real number \(x\) for which \( \sqrt{-x} \) is less than 0. Therefore, the inequality \( \sqrt{-x} < 0 \) has no solution in the real numbers because a square root cannot be negative.

Step 4: Conclusion

Since no real number can satisfy \( \sqrt{-x} < 0 \), the inequality has no solution over the real numbers.

Key Concepts

Real Numbers

Real numbers are a foundational concept in mathematics. They include all the numbers we typically use in everyday life and calculations, such as integers, fractions, and decimals. Real numbers are part of a vast number system that comprises both rational and irrational numbers.

• Rational numbers are those that can be expressed as a fraction of two integers, like \( \frac{1}{2} \) or 3.
• Irrational numbers cannot be expressed as fractions; a common example is \( \pi \) or the square root of 2 (\( \sqrt{2} \)).

Real numbers can be represented on a continuous number line, extending infinitely in both positive and negative directions. For inequalities, like the one in the original exercise, understanding real numbers is crucial, as it determines the possible values that \( x \) can take. In our exercise, no real number \( x \) will satisfy \( \sqrt{-x} < 0 \) because real numbers do not include imaginary numbers, which would be required for taking the square root of a negative number.

Square Roots

Understanding square roots is key to solving many mathematical problems involving inequalities. A square root of a number \( a \) is a value that, when multiplied by itself, gives \( a \). The most common square root symbol is \( \sqrt{} \).

• The square root of a positive number is straightforward, e.g., \( \sqrt{16} = 4 \), because \( 4 \times 4 = 16 \).
• For zero, \( \sqrt{0} = 0 \), as anything multiplied by zero is 0.
• For negative numbers, square roots are not real numbers since no real number multiplied by itself gives a negative. Instead, this involves imaginary numbers, noted with the symbol \( i \), where \( i^2 = -1 \).

In our exercise, \( \sqrt{-x} \) implies we would take the square root of a negative if \( x \) were positive, which is not possible in the set of real numbers. This constraint helps us explore the inequality's solution properly.

Domain Identification

Domain identification is an essential first step in solving inequalities, especially those involving square roots. The domain defines all possible input values ("\( x \)-values") for which a mathematical expression is valid.

• For square roots, the interior of the \( \sqrt{} \) symbol must be non-negative for the expression to remain in the real numbers.
• For the expression \( \sqrt{-x} \), this means \(-x\geq 0\), or simply \( x\leq 0 \).
• Identifying the domain helps prevent calculations with undefined expressions; in this exercise, that means avoiding attempts to find a real square root of a negative number.

For the inequality \( \sqrt{-x} < 0 \), identifying that \( x \) must be \( \leq 0 \) narrows the range of real numbers to consider. However, even within this domain, \( \sqrt{-x} \) cannot be less than zero, leading to the conclusion that within the domain of real numbers, there is no \( x \) for which the inequality holds true.