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 that are both 7 units away from -5 on a number line.

Draw a Number Line

Visualize a number line with -5 marked on it. Our task is to find two points, one on either side of -5, that are 7 units away.

Calculate the Point to the Right

To find the point 7 units to the right of -5, add 7 to -5:\(-5 + 7 = 2\). So, one point is 2.

Calculate the Point to the Left

To find the point 7 units to the left of -5, subtract 7 from -5:\(-5 - 7 = -12\). So, the other point is -12.

Key concepts

Distance on Number Line

Distance on the number line is a measure of how far apart two points are. To visualize this, imagine a straight line with numbers placed at equal intervals. Each number represents a point, and the spaces between them illustrate the units of distance.

When we say two points are a certain number of units apart, we mean the total number of steps or intervals between them on this line. In our exercise, the problem asks to find points that are 7 units away from (-5").

The number (-5") would be positioned on the number line, and you are tasked with finding both a point 7 steps to the right and one 7 steps to the left. These steps are helpful to determine the potential locations where these points might exist. The concept of distance becomes very intuitive on the number line, as you can visually see where these steps land.

Number Line Addition

Number line addition is simply moving to the right on the number line. When you add a positive number to another number, you step forward by that number of units. This is helpful for visual learners to grasp addition as movement rather than abstract numbers.

Let's see this in the context of the exercise. Starting from (-5"): you add 7, moving 7 steps right. In simple terms, (-5 + 7"), moving from (-5"), brings you to 2.

This is a straightforward way of understanding that adding numbers on a line equates to finding their placement on that line. This creates a simple yet effective visual learning tool, especially beneficial for students struggling with abstract mathematical concepts. Remember, each addition on the line is just like taking a walk! You move forward in the positive direction.

Number Line Subtraction

Number line subtraction involves moving to the left on the number line. When you subtract a number from another, imagine taking steps backward by the number of units you are subtracting.

In our problem, to find a point that is 7 units away from (-5"): we subtracted. Starting from (-5") and subtracting 7, (-5 - 7"). This results in (-12").

Visual learners find it helpful to see subtraction as a reverse journey, moving leftwards into negative territory if necessary. Breaking it down: (-5 - 7") is just stepping back 7 units from (-5"), leading us to (-12").

This visual approach makes understanding subtraction simpler and consistent with everyday experiences, where backward movement reduces numbers or value. Subtraction is made clearer with the number line, depicting declining value as steps to the left.