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

Understand the Problem

We need to find two points on a number line that are exactly 7 units away from the point \(-5\). This involves moving 7 units to the right and 7 units to the left from \(-5\).

Calculate Rightward Point

Start at \(-5\) and move 7 units to the right: \(-5 + 7 = 2\). So, one point is at 2 on the number line.

Calculate Leftward Point

Start at \(-5\) and move 7 units to the left: \(-5 - 7 = -12\). So, the second point is at \(-12\) on the number line.

Verification

Verify that both points are indeed 7 units away from \(-5\). For 2, the distance is \(|2 - (-5)| = 7\). For -12, the distance is \(|-12 - (-5)| = 7\). Both are correct.

Key concepts

Understanding Distance on the Number Line

The concept of distance on a number line is central to understanding math problems involving points on a line. On a number line, distance is the number of units between two points, regardless of direction. Even if a number is negative, like (-5), the distance remains a positive number. Distance doesn't care about negative numbers; it just counts how far you travel on the line. To find distance between two points, you subtract the smaller number from the larger number and take the absolute value of the result. Distance, therefore, is always non-negative. For example:

• Between (-5) and 2: |2 - (-5)| = |2 + 5| = 7
• Between (-5) and (-12): |-12 - (-5)| = |-12 + 5| = 7

So, whether you go right or left from a point, like (-5), the distance is simply about how many steps you take on the number line.

The Role of Negative Numbers

Negative numbers can seem tricky, but they're just numbers on the number line! Think of them as steps to the left of zero. When you add to a negative number, you're moving right on the number line. Conversely, subtracting from a negative number means stepping even more to the left. This behavior accounts for why (-5) + 7 brings us to 2, while (-5) - 7 takes us to (-12). The number line extends infinitely in both directions. Therefore, negative numbers are just as valid as positive ones.

• Adding to a negative: moves right
• Subtracting from a negative: moves left

Using negative numbers correctly helps you pinpoint precise locations on the line, whether it is for distances or other calculations.

Enhancing Critical Thinking with Number Lines

Critical thinking in math means breaking down problems into more straightforward steps. Number lines are excellent tools for this because they visually show relationships between numbers. First, understand what the problem is asking. In our example, we needed to find two points relative to (-5) with a fixed distance. Then, use methodical steps to solve it:

• Identify what operation to perform (addition or subtraction).
• Visualize moving left or right on the line.
• Check your work using distances.

Critical thinking encourages us not just to calculate but to verify and understand why our answers are correct. When you see a number line, don't just see numbers. Instead, think of it as a map that guides you through problems with confidence. By practicing these steps, we're also improving problem-solving skills in everyday life!