Problem 95 - Elementary Algebra

Question

Fill in the blank with \(<,=\), or \(>\). -7 _____ \(-(-8)\)

Step-by-Step Solution

Answer

-7 < 8.

Step 1: Analyze Numbers

The first number given is \(-7\). It is clearly a negative number. The second number is \(-(-8)\), which might seem complicated initially. We need to simplify \(-(-8)\) to understand its value.

Step 2: Simplify Second Number

The expression \(-(-8)\) refers to the negative of a negative number. When we negate a negative number, it becomes positive. Therefore, \(-(-8) = 8\).

Step 3: Compare Values

Now we compare the simplified values of \(-7\) and \(8\). Since \(-7\) is a negative number and out of the two, \(8\) is positive, clearly, \(-7\) is less than \(8\).

Step 4: Fill in the Blank

Since \(-7 < 8\), we fill in the blank with \(<\) to make the statement true.

Key Concepts

Negative Numbers

Negative numbers are those numbers that are less than zero. They are expressed with a minus sign (-) in front of the number. These numbers are found to the left of zero on a number line. For instance,

• -7
• -1
• -20

Negative numbers can represent several real-world situations like debt, temperature below freezing, or altitude below sea level.

When comparing negative numbers, it is important to remember that the number closer to zero is greater. So,

• -5 is greater than -10 because -5 is closer to zero.
• -3 is greater than -7.
• -1 is greater than -8.

An interesting property of negative numbers is that when you add or subtract them, the operation influences their direction on the number line. If you add a negative number, you move left. Subtract a negative, and you move right, which is essentially adding a positive number.

Positive Numbers

Positive numbers are numbers greater than zero and are usually written without any sign or with a plus sign (+). These numbers appear to the right of zero on a number line, like so:

• 1
• 7
• 15
• 52

Positive numbers are used frequently in real-life contexts, such as counting objects, measuring distances, and representing gains or profits.

When comparing positive numbers, the number further from zero is the greater one. For example:

• 9 is greater than 5 because 9 is further from zero than 5.
• 22 is greater than 18.
• 100 is greater than 50.

In addition, when operations such as addition, subtraction, multiplication, and division are performed with positive numbers, they typically yield another positive number, unless division by a larger positive number results in a smaller fraction.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form to make them easier to understand and solve. This process often involves combining like terms, reducing expressions, or handling multi-step operations.

For instance, with the expression \[-(-8)\] the inner negative sign is applied to 8, and then the outer negative negates it again. This results in a positive 8 because \[-(-8) = 8.\]This example demonstrates that a negative of a negative number results in a positive number.

Simplifying different types of expressions means:

• Combining like terms: \(2x + 3x = 5x\).
• Reducing fractions: \(\frac{8}{12} = \frac{2}{3}\).
• Using the distributive property: \(2(a + b) = 2a + 2b\).

Applying these steps helps in solving complex problems efficiently. Simplification is fundamental in algebra and helps in comparing different expressions and solving equations easily.