Problem 3 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 3-6: Replace the symbol \(\square\) with either \(<,>\), or \(=\) to make the resulting statement true. (a) \(-7 \square-4\) (b) \(\frac{\pi}{2} \square 1.57\) (c) \(\sqrt{225} \square 15\)

Step-by-Step Solution

Answer

(a) <; (b) =; (c) =

Step 1: Determine the Relationship for (a)

We need to compare the two numbers: - Left side: \(-7\)- Right side: \(-4\)Since \(-7\) is less than \(-4\), replace the symbol \(\square\) with \(<\).

Step 2: Assess the Value of \(\pi/2\) for (b)

First, calculate the approximate value of \(\frac{\pi}{2}\). We know \(\pi \approx 3.14\), so \(\frac{3.14}{2} \approx 1.57\). Hence, the expression \(\frac{\pi}{2} = 1.57\). Replace \(\square\) with \(=\).

Step 3: Calculate the Square Root for (c)

Find the square root of 225. Since \(15^2 = 225\), it follows that \(\sqrt{225} = 15\). Therefore, the expression should be replaced with an \(=\) symbol.

Key Concepts

Inequalities

Inequalities are used to compare two values or expressions, showing which is greater, lesser, or if they are equal. In arithmetic, we often use symbols like \(<\), \(>\), and \(=\). Understanding inequalities helps in solving problems where precise relationships need to be determined.
For example, in comparing

• -7 and -4, we see that the position on the number line matters.
• -7 lies leftward, indicating it is less than -4.

Hence, the appropriate symbol is \(<\). It is essential to remember that a smaller negative number is further from zero, making it less than a larger negative number.

Numeric Approximations

Numeric approximations are often used when exact values like \(\pi\) need to be represented in simpler terms for calculations. Since \(\pi\) is an irrational number with a never-ending decimal expansion, we can approximate it for ease of calculations.
For practical purposes,

• we often use 3.14 as an approximation for \(\pi\).
• In the absence of a calculator, dividing 3.14 by 2 gives approximately 1.57.

This way, \(\frac{\pi}{2} \approx 1.57\). With this approximation, comparing \(\frac{\pi}{2}\) and 1.57, the use of \(=\) is accurate enough because 1.57 is very close to the true value of \(\frac{\pi}{2}\). Approximations are useful as they simplify problems without significantly affecting the accuracy of results.

Square Roots

Square roots provide a way to find a number which, when multiplied by itself, equals a given number. This concept is crucial for understanding how numbers relate to each other in terms of area and geometry.
Taking the example of:

• The number 225, where its square root is the number that gives 225 when multiplied by itself.
• Since \(15^2 = 225\), \(\sqrt{225} = 15\).

Thus the symbol \(=\) is used here, indicating that they are equal. Understanding square roots helps solve problems involving quadratic equations, geometry, and is foundational for higher-level mathematical concepts.