Problem 59 - Elementary Algebra

Question

True or false. $$ -1000<-20 $$

Step-by-Step Solution

Answer

True.

Step 1: Understand the Inequality Symbols

The inequality symbol '<' means 'is less than'. So, the statement \(-1000 < -20\) is asking if -1000 is less than -20.

Step 2: Visualize the Number Line

On a number line, numbers to the left are smaller than numbers on the right. -1000 is placed far to the left of -20 because it is a larger negative number. This means -1000 is less than -20.

Step 3: Analyze the Statement

Since -1000 is to the left of -20 on the number line, this confirms that \(-1000 < -20\) is true.

Key Concepts

Negative Numbers

When dealing with numbers that have a minus sign in front of them, we call them negative numbers. They're the opposite of positive numbers and always have a value less than zero. Imagine these numbers as steps back from zero on the number line.

• Negative numbers are denoted with a '-' sign, such as -2, -5, or -1000.
• The smaller the number is (in positive value), the larger it is as a negative—it might sound confusing but think of it in terms of distance from zero.

Understanding negative numbers helps when comparing values. For instance, -1000 and -20 are both negative, but because -1000 is further from zero, it's actually smaller than -20. This is a key concept in solving inequalities with negative numbers.

Number Line

A number line is a straight line with numbers placed at equal intervals or segments along its length. It can be a handy tool to visualize and compare numbers, especially when they are both negative.

• Numbers decrease as you move left along the number line.
• Conversely, numbers increase as you move to the right.

When you picture \(-1000\) and \(-20\) on a number line, it helps to remember that any number further to the left is smaller. So, in this case, -1000 appears far to the left of -20, confirming that it is indeed smaller when we're talking about negative numbers.

Inequality Symbols

Inequality symbols are mathematical symbols that show the relationship between two values. They tell us whether one value is larger or smaller than another. The typical symbols include:

• '<' which means 'less than'.
• '>' which indicates 'greater than'.
• '≤' which stands for 'less than or equal to'.
• '≥', meaning 'greater than or equal to'.

In an inequality like \(-1000 < -20\), the symbol '<' tells us to compare the two numbers. In this case, the statement asks if -1000 is smaller than -20. By using our understanding of the number line and negative numbers, we see that this is true, showing the power and flexibility of inequality symbols in math.