Question

Critical Thinking Name two points that are 7 units from \(-5\) on the number line. (Hint: Use a number line.)

Short answer

The points are 2 and -12.

Step-by-step solution

Understanding the problem

We are tasked with finding two points on the number line that are each 7 units away from the point \(-5\). This means moving 7 units to the right and 7 units to the left from the point \(-5\).

Moving 7 units to the right

Starting at \(-5\), to find the point 7 units to the right, add 7 to \(-5\):\[-5 + 7 = 2\]This calculation shows that one point that is 7 units from \(-5\) is \(2\).

Moving 7 units to the left

Starting again at \(-5\), to find the point 7 units to the left, subtract 7 from \(-5\):\[-5 - 7 = -12\]This calculation indicates that the second point that is 7 units from \(-5\) is \(-12\).

Conclusion

After calculating both directions, we can conclude that the two points that are 7 units away from \(-5\) on the number line are \(2\) and \(-12\).

Key concepts

Distance on a Number Line

Think of the number line as a simple but powerful tool that helps visualize distances between points. Each number on the line represents a point. The distance between these points is measured in units along the line itself. Imagine the number line as a straight pathway where every tick mark is a step or a unit. The distance is simply the count of these steps between two points. In our exercise, the task was to find points 7 units away from \(-5\). This involves moving to the right or left along the line by 7 steps:

• To the right of \(-5\) means adding 7 units, arriving at \(2\).
• To the left of \(-5\) involves subtracting 7 units, reaching \(-12\).

Number lines make understanding such shifts intuitive, as movements along them are visually clear.

Integers

Integers are whole numbers that can be positive, negative, or zero. They do not include fractions or decimals, and they extend infinitely in both directions on the number line.Understanding integers is key to math because they represent various quantities in real-life situations, like gains, losses, temperatures, or elevations. In the context of a number line:

• Positive integers are located to the right of zero.
• Negative integers reside to the left of zero.
• Zero is the neutral point directly in the middle.

In our exercise, \(-5\), \(2\), and \(-12\) are all integers. Using integers allows us to easily perform operations like addition and subtraction to determine distance.

Addition and Subtraction

Addition and subtraction are fundamental operations in math that allow us to calculate distances, especially on a number line.

• **Addition** involves combining numbers to find a total. On a number line, adding a positive number moves you right, increasing the value.
• **Subtraction** is about taking one number away from another. Subtracting a positive number moves you left, decreasing the value.

For our exercise, adding 7 to \(-5\) yielded \(2\), a process that moves right on the number line. Conversely, subtracting 7 from \(-5\) gave \(-12\), moving left. These operations are straightforward yet powerful, offering a clear method to find distances and understand positions on a number line.