Problem 29 - Prealgebra

Question

Replace each \(\odot\) with \(<,>,\) or \(=\) to make a true statement. $$-\sqrt{6.25} \odot-\frac{5}{2}$$

Step-by-Step Solution

Answer

Use "=" because both sides are equal, both being -2.5.

Step 1: Simplify the Square Root

First, simplify the square root expression. Calculate \(\sqrt{6.25}\), which is \(2.5\). Therefore, \(-\sqrt{6.25} = -2.5\).

Step 2: Simplify the Fraction

Next, simplify the fraction \(-\frac{5}{2}\). Divide 5 by 2 to get 2.5. Multiply by -1 to get \(-2.5\).

Step 3: Compare the Values

Now compare \(-2.5\) and \(-2.5\). Since both values are equal, the correct symbol to use is \(=\).

Key Concepts

Understanding Square Roots

A square root is a number that, when multiplied by itself, gives you the original number. So, if you have a number like 6.25 and you want to find its square root, you are looking for a number which when squared (multiplied by itself) equals 6.25. In this case, that number is 2.5 because \(2.5 \times 2.5 = 6.25\).

The square root of a number can have two values: one positive and one negative. For example, the square root of 4 can be both 2 and -2, since \(2 \times 2 = 4\) and \(-2 \times -2 = 4\). Generally, unless specified, we take the positive square root in math problems.

When dealing with square roots in equations or comparisons, you may first want to calculate the square root value to simplify the expression, just like we did for the exercise where \(-\sqrt{6.25}\) was simplified to \(-2.5\). Understanding the step of simplifying is crucial in comparing numbers, especially when they involve other operations such as addition or negative signs.

Breaking Down Fractions

Fractions represent parts of a whole. In the expression \(-\frac{5}{2}\), the number above the line (numerator) is 5, and the number below the line (denominator) is 2. This fraction shows 5 divided by 2. When you divide 5 by 2, you get 2.5. The negative sign in front indicates it is below zero, so the fraction simplifies to \(-2.5\).

To simplify a fraction, try dividing the numerator by the denominator. This gives you a decimal or whole number value that makes problems easier to solve. It's particularly helpful when comparing numbers, as decimals show the magnitude directly. Always remember, a negative sign in front of a fraction indicates the result will also be negative, which plays a key role when you are comparing this with other numbers—especially when comparing with negative square roots.

The Role of Negative Numbers

Negative numbers are numbers less than zero, represented with a minus sign \(-\). When you have an expression like \(-\sqrt{6.25}\) or \(-\frac{5}{2}\), you deal with negative values. Negative numbers have unique properties:

• Adding a negative number is like subtracting its positive counterpart.
• Comparing two negative numbers can be tricky; the number with the smaller absolute value is actually larger.
• Multiplying two negative numbers results in a positive product.

For example, when you compare \(-2.5\) and \(-2.5\), since their values are equal, you determine that neither is greater, thus \(=\) is the appropriate symbol. Understanding how negative numbers work helps in solving equations and comparing numbers, crucial for dealing with various math problems.