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 need to find two points on the number line that are exactly 7 units away from a given point, which is -5.

Use of Absolute Value

Since distance is always positive, we utilize the absolute value concept. For a point to be 7 units away from -5, we set up the equation: \(|x + 5| = 7\), where \(x\) is the unknown point.

Setting up Equations

From the absolute value equation \(|x + 5| = 7\), we derive two cases: \(x + 5 = 7\) and \(x + 5 = -7\).

Solving Case 1

For the first case \(x + 5 = 7\), solve for \(x\): \(x = 7 - 5\). Thus, \(x = 2\).

Solving Case 2

For the second case \(x + 5 = -7\), solve for \(x\): \(x = -7 - 5\). Thus, \(x = -12\).

Conclusion

The two points that are 7 units away from -5 on the number line are \(x = 2\) and \(x = -12\).

Key concepts

Distance on a Number Line

The concept of distance on a number line is foundational. It refers to the length between two points. For example, consider the point \(-5\). The goal is to find two points 7 units away from \(-5\). **Distance**, in this context, is always measured as a positive number, meaning you count spaces rather than direction. It's like seeing how many steps you need to take to get from one point to another.

Think of the number line as a ruler stretching infinitely in both directions. When assessing distance, the direction left or right is equally probable. To be precise, you need to count 7 units from \(-5\) in both directions. This will give two potential solutions: \(-5 + 7\) and \(-5 - 7\). Each counts the same number of spaces, but in opposite directions.

This demonstrates that there are always two points that maintain a specific distance from a reference point on a number line, one to the left and one to the right. Highlighting this aspect facilitates understanding of absolute values and reinforces symmetrical properties of numbers on the line.

Solving Equations

Solving equations is a systematic method for finding solutions. In the exercise, you solve for points 7 units away from \(-5\) using the concept of absolute value. Start with the equation \(|x + 5| = 7\). An absolute value equation can be tackled by considering both positive and negative solutions.

Here is how you proceed:

• First, remove the absolute value by setting up two separate equations: \(x + 5 = 7\) and \(x + 5 = -7\).
• The first case \(x + 5 = 7\) implies moving to the right. Solve this by subtracting 5 from both sides, resulting in \(x = 2\).
• In the second case \(x + 5 = -7\), you shift left. By subtracting 5 from -7, find \(x = -12\).

These solutions, \(x = 2\) and \(x = -12\), maintain a distance of 7 units from \(-5\) as required in the problem.

Absolute value equations often lead to two solutions because they account for symmetrically opposite yet valid scenarios. This principle is essential for understanding broader mathematical concepts, revealing the symmetry in numbers.

Critical Thinking in Mathematics

Critical thinking is crucial in math, enabling students to solve problems efficiently. In the context of this exercise, it involves a deep understanding of the number line and the application of absolute values. Critical thinking helps break down complex problems into manageable steps.

Begin by identifying what the problem asks; in this case, finding points at a set distance from a number. Recognize that distance involves absolute values, which inform the equations you'll form.

Whenever you tackle such problems, consider:

• **Understanding the Problem:** Determine what information you have and what is sought.
• **Formulating a Strategy:** Use known formulas and operations, like absolute value operations, to guide your problem-solving process.
• **Solution Verification:** Check if the solutions make sense within the context, such as verifying that both points indeed lie the required distance from the initial point on the number line.

Practicing this habit enhances mathematical proficiency and allows for confident navigation through various math challenges. Each time you solve a problem, think about the methods involved and how alternative paths may arrive at the same conclusion.