Problem 21 - Algebra 1

Question

Graph the numbers on a number line. Then write two inequalities that compare the two numbers. $$-0.1 \text { and }-0.11$$

Step-by-Step Solution

Answer

\(-0.1 > -0.11 \) and \(-0.11 < -0.1 \)

Step 1: Plot the numbers on a number line

To do this, draw a horizontal line, and mark off evenly spaced intervals. For this case, because we are dealing with decimals, it might be easier to plot increments of 0.01. Mark -0.1 and -0.11 on this line. Since -0.1 is greater than -0.11, it should be located to the right of -0.11 on a properly oriented number line.

Step 2: Write the first inequality

An inequality is a relation between two expressions that may not be equal. Based on the graphic representation on the number line, -0.1 is greater than -0.11. This can be written as -0.1 > -0.11.

Step 3: Write the second inequality

Similarly, the other inequality will express that -0.11 is less than -0.1. This can be written as -0.11 < -0.1.

Key Concepts

Number Line

A number line is a helpful visual tool in mathematics that allows us to better understand the relationship between numbers, especially when comparing them. Imagine a horizontal line with evenly spaced increments. This line continues infinitely in both directions, with zero as a central reference point. Numbers to the right of zero are positive, and numbers to the left are negative.

When you plot numbers like \(-0.1\) and \(-0.11\) on a number line, they are placed according to their value relative to zero and each other. A key point to remember is:

• Negative numbers closer to zero are greater than those further away.
• Each decimal place represents a finer division on the line; thus, \(-0.1\) is located to the right of \(-0.11\).

This visualization aids in forming and understanding inequalities by showing how the numbers relate spatially.

Decimals

Decimals are a way to represent fractions and real numbers with a point ("decimal point") that separates the whole part from the fractional part. Understanding decimals is crucial when dealing with precise values and measurements, like when comparing \(-0.1\) and \(-0.11\).

It's important to note:

• Each digit to the right of a decimal point represents a power of ten. For example, in \(-0.1\), the "1" is in the tenths place, and in \(-0.11\), the first "1" is in the tenths place and the second "1" is in the hundredths place.
• Being familiar with decimals helps in accurately plotting them on number lines and making comparisons easier.

When converting decimals into a number line format, the place value significantly affects their positioning.

Graphing Inequalities

Graphing inequalities involves plotting numbers on a number line and determining which numbers satisfy the inequality. An inequality compares two values, indicating whether one is greater than, less than, or possibly equal to another. For example, when analyzing \(-0.1 > -0.11\), the number line is crucial: it shows \(-0.1\) to the right of \(-0.11\), supporting this inequality.

Remember these steps:

• Identify and plot the numbers on a number line.
• Use symbols like ">" and "<" to express the inequality between numbers.
• The position of numbers on the line visually represents their relationship.

Graphing inequalities is a useful method for understanding the relative sizes of numbers, bringing a visual element to abstract numerical concepts.