Problem 51 - Algebra 1

Question

Graph the numbers on a number line. Then write two inequalities that compare the two numbers. $$-0.2 \text { and }-0.21$$

Step-by-Step Solution

Answer

-0.21 < -0.2 and -0.21 ≤ -0.2

Step 1: Plot the numbers on number line

To graph -0.2 and -0.21 on a number line, start by drawing a straight horizontal line. Mark '0' somewhere in the middle. Then, mark negative and positive numbers on both sides of '0' with equal spacing. The distance between every two consecutive numbers should be the same. Plot -0.2 somewhere to the left of '0' (because it's a negative number), and then plot -0.21 a little bit more to the left of -0.2.

Step 2: Compare the numbers

Once the numbers are graphed, notice that -0.21 falls to the left of -0.2 on the number line. In number line, left means lesser and right means greater. Therefore, -0.21 is less than -0.2.

Step 3: Write two inequalities

Based on the comparison we have made, now write two inequalities. The first inequality is without the equation sign, i.e. -0.21 < -0.2. The second inequality is with the equation sign, i.e. -0.21 ≤ -0.2. The first inequality signifies that -0.21 is strictly less than -0.2 while the second one means -0.21 is less than or equal to -0.2.

Key Concepts

Number Line

A number line is a visual representation of numbers on a straight line, which makes it easier to understand concepts like inequalities, addition, and subtraction. The number line is marked with numbers at regular intervals, typically including both negative, positive, and zero.
One of the key things to remember about the number line is:

• Positive numbers are placed to the right of zero.
• Negative numbers are placed to the left of zero.
• The further left you go means the numbers are smaller, while further right means they are larger.

In exercises that involve plotting decimals, such as \(-0.2\) and \(-0.21\), it's important to find the correct position between whole numbers. This can be tricky, so let's think of the number line like a ruler - dividing the spaces between whole numbers accurately will help you place the decimals in the right spot.

Negative Numbers

Negative numbers are values less than zero, indicated by a minus sign (\(-\)). They can often cause confusion, especially when comparing two negative numbers. Here's a fun fact about negative numbers:

• In finance, negative numbers are often used to indicate a loss or debit.
• In temperature, negative values represent temperatures below freezing.

When comparing negative numbers, it’s important to remember that the number further to the left on a number line is smaller. For example, on the number line, \(-0.21\) is to the left of \(-0.2\), making \(-0.21\) the smaller number. This is opposite of what we see with positive numbers. This is because negative numbers represent a lack of value compared to zero and positive numbers.

Graphing Numbers

Graphing numbers on a number line allows for a clear and visual method of comparing numbers. To graph numbers like \(-0.2\) and \(-0.21\), follow these steps:

• Draw a horizontal line.
• Mark zero towards the middle, then label negative values to the left and positive values to the right.
• Locate precise points for decimal numbers between whole numbers. For this, imagine the space between each whole number divided into equal parts.

Graphing shows the relative position of numbers and helps in quickly identifying which number is larger or smaller, thanks to the position on the line. For inequalities, such as comparing \(-0.21\) and \(-0.2\), seeing these numbers on the number line makes it clear that \(-0.21\) is less than \(-0.2\), supporting mathematical expressions like \(-0.21 < -0.2\) and \(-0.21 \leq -0.2\). This visual tool simplifies understanding of complex concepts like inequalities.