Problem 73 - Algebra 1: Concepts and Skills

Question

Graph the numbers on a number line. Then write two inequalities that compare the numbers. \begin{equation} 0.4,-3 \end{equation}

Step-by-Step Solution

Answer

The numbers -3 and 0.4 can be plotted on a number line, which allows the writing of the following two inequalities: -3<0.4 and 0.4>-3.

Step 1: Plotting the numbers

For this step, a number line is needed. The zero point is marked first, then the numbers \$0.4\$ and \-3\ are plotted on this number line. As \-3\ is less than zero, it goes to the left of zero. Whereas, \$0.4\$ is greater than zero, so it is plotted to the right of zero.

Step 2: Writing the inequalities

Once the numbers are plotted, their relative positions on the number line can be used to write the inequalities. In this case, because the point representing -3 is to the left of 0.4, it follows that \-3\ is less than 0.4. We can therefore write the inequalities as \-3<0.4\ and \$0.4> -3\$.

Key Concepts

Understanding the Number Line

A number line is a visual representation of numbers laid out on a straight, horizontal line. It helps us easily see the order and spacing of numbers. Typically, zero is placed in the middle, with positive numbers extending to the right and negative numbers extending to the left.

The number line is useful because it provides clarity on where numbers are positioned relative to each other. By visualizing numbers in this way, it becomes easier to compare them and see relationships between different values.

When working with exercises involving the number line, always start by identifying significant points like zero, positive increments, and negative increments.

Plotting Numbers Accurately

When plotting numbers on a number line, it's essential to place them accurately to reflect their true value compared to others. Start by marking the zero point for reference. Then proceed to place other numbers based on their value relative to zero.

For instance, if you’re plotting the numbers 0.4 and -3:

• - Zero is typically marked at the center of the number line.
• - Number 0.4, being a positive number, should be placed to the right of zero.
• - Number -3, being a negative number, should be placed to the left of zero.

This accurate plotting helps prevent mistakes in interpreting their values and comparing them effectively in the next steps.

Comparing Numbers Using Inequalities

After plotting numbers on a number line, it's straightforward to compare them because their positions clearly indicate which is greater or lesser.

Inequalities are symbols that express the relationship between two values. In this context:

• "<" means "less than"
• ">" means "greater than"

From the exercise, once -3 is plotted to the left of 0.4, it visually confirms that -3 is less than 0.4.

Consequently, we can write the inequalities as \(-3 < 0.4\) and \(0.4 > -3\). This method of comparison is vital in mathematics as it provides a clear and accurate way to express the relationship between numbers based on their positions on the number line.