Problem 67 - Algebra and Trigonometry

Question

\(67-70=\) Determine the values of the variable for which the expression is defined as a real number. $$ \sqrt{16-9 x^{2}} $$

Step-by-Step Solution

Answer

\([-\frac{4}{3}, \frac{4}{3}]\) is the solution set for real numbers.

Step 1: Understand the Square Root Condition

For the square root \( \sqrt{16-9x^2} \) to be defined as a real number, the expression inside the square root must be greater than or equal to zero. This means we need to solve the inequality \( 16 - 9x^2 \geq 0 \).

Step 2: Rearrange the Inequality

Rearrange the inequality to \( 16 \geq 9x^2 \). This will make it easier to isolate \( x \).

Step 3: Divide and Simplify

Divide each side of the inequality \( 16 \geq 9x^2 \) by 9 to simplify it: \( \frac{16}{9} \geq x^2 \).

Step 4: Solve for \( x \)

To remove the square, take the square root of both sides. Remember to consider both the positive and negative square roots: \( -\frac{4}{3} \leq x \leq \frac{4}{3} \).

Step 5: Solution Set

The values of \( x \) that satisfy this condition form the interval \([-\frac{4}{3}, \frac{4}{3}]\).

Key Concepts

Square Root Function

The square root function is an essential concept in mathematics, often denoted by the radical symbol \( \sqrt{} \). When you see \( \sqrt{a} \), it asks for a number which, when multiplied by itself, gives you \( a \). For example, \( \sqrt{9} = 3 \) because \( 3 \times 3 = 9 \).

However, it's crucial to remember that a square root function dealing with real numbers has specific conditions. The expression inside the square root, which we call the radicand, must be non-negative, meaning it needs to be zero or positive. That's because square roots of negative numbers aren't real numbers (they're imaginary numbers).

• If you have \( \sqrt{16-9x^2} \), it means the radicand \( 16-9x^2 \) must be \( \geq 0 \).
• This requirement is fundamental for ensuring the result of the square root function remains within the realm of real numbers.

Paying attention to these conditions is essential when solving inequalities involving square root functions.

Inequalities

Inequalities allow us to express a range of solutions rather than a specific number. They are written using symbols like \( >, <, \geq, \leq \). Here, we are dealing with an inequality that ensures the radicand of the square root function is non-negative: \( 16 - 9x^2 \geq 0 \).

To solve inequalities:

• First, rearrange the expression so you can isolate the variable. For \( 16 - 9x^2 \), we rearrange it to \( 16 \geq 9x^2 \).
• Then, simplify the inequality by dividing, if needed. Here, we divide both sides by 9, giving us \( \frac{16}{9} \geq x^2 \).
• Finally, solve for \( x \). In this case, you find the roots by taking the square root of each side: \(-\frac{4}{3} \leq x \leq \frac{4}{3} \).

This sequence allows us to find out which values of \( x \) make the original expression meet the necessary condition for being a real number.

Interval Notation

Interval notation is a compact and efficient way of representing a range of numbers. In our solution, we determined the values for \( x \) that satisfy the inequality are between \(-\frac{4}{3}\) and \(\frac{4}{3}\).

In interval notation:

• The interval \([a, b]\), where \(a\) and \(b\) are numbers, includes all numbers between \(a\) and \(b\), inclusive. The square brackets \([\ ]\) denote that \(a\) and \(b\) are included within the interval.
• In our example, \([-\frac{4}{3}, \frac{4}{3}]\) means that \(x\) can be any value between \(-\frac{4}{3}\) and \(\frac{4}{3}\), including \(-\frac{4}{3}\) and \(\frac{4}{3}\) themselves.

Understanding how to write and interpret interval notation ensures clear communication of which values satisfy conditions like the one found when working with inequalities in mathematics.