Problem 120 - Contemporary Precalculus

Question

Translate the given algebraic statement into a geometric statement about numbers on the number line. $$|u+v| \geq 2$$

Step-by-Step Solution

Answer

Question: Translate the algebraic statement "|u+v| ≥ 2" into a geometric statement about the positions of u and v on a number line. Answer: The sum of two numbers u and v on the number line must be located at least 2 units away from 0, either to the right or to the left.

Step 1: Understanding Absolute Value

Absolute value is the distance of a number from zero on the number line. It is denoted by ||. In this case, we are dealing with a sum of two numbers, u + v, enclosed in absolute value symbols: |u+v|. Hence, we need to find the distance between the result of this sum and the zero point on the number line.

Step 2: Understanding the Inequality

The given inequality is: $$|u+v| \geq 2$$ This means that the distance between the sum of the two numbers u and v from zero on the number line is greater than or equal to 2.

Step 3: Defining the Geometric Statement

If the distance between the sum (u + v) and 0 is greater than or equal to 2, this means that the sum of u and v must lie either to the right of the number 2 on the number line, or to the left of the number -2 on the number line. Since absolute values focus on the distance from the zero point, the resulting geometric statement would be that the sum of u and v must be positioned at least 2 units away from 0, in either direction. This can be represented on the number line with all the possible areas where the sum of u and v could lay. In summary, the geometric statement for the given algebraic statement is: The sum of two numbers u and v on the number line must be located at least 2 units away from 0, either to the right or to the left.

Key Concepts

Number Line

The number line is a simple yet powerful tool in mathematics that helps us visually represent numbers and understand their relationships with one another. Think of it as a straight line where each point corresponds to a real number. The center of this line is marked as zero. To the right of zero, positive numbers are placed, increasing in value as you move further right. To the left of zero, negative numbers reside, decreasing in value as you move to the left.

When dealing with absolute values, the number line becomes incredibly helpful. Absolute value measures the distance a number is from zero on this line, regardless of direction. For instance, both +3 and -3 have an absolute value of 3 because they are both three units away from zero. When translating algebraic expressions like \(|u+v|\), the number line clearly illustrates that the sum of values \(u+v\) should maintain a certain distance from zero.

Inequality

Inequalities are statements about the relative size or order of two values. They tell us whether one number is larger, smaller, or possibly equal to another. The inequality \(|u+v| \geq 2\) states that the absolute value of the sum of \(u\) and \(v\) is greater than or equal to 2.

This means that regardless of the direction on the number line, the sum must be at least 2 units away from zero. Inequalities use symbols like \(>\), \(<\), \(\geq\), and \(\leq\) to convey these relationships. Here, the symbol \(\geq\) means "greater than or equal to" which introduces more possibilities than a strict "greater than" might. In our scenario, \(u+v\) could be exactly 2 units away or more.

Geometric Translation

Geometric translation involves representing algebraic statements through visual, geometric interpretations. It acts as a bridge between algebra and geometry, allowing us to visualize abstract algebraic concepts on a number line.

In translating the inequality \(|u+v| \geq 2\) into a geometric statement, we express it as the location of \(u+v\) on the number line. When we say \(|u+v| \geq 2\), we are indicating that \(u+v\) cannot be between -2 and 2. Instead, it must lie outside these bounds, either less than -2 or greater than 2. This gives us a clear picture: imagine shading regions on the number line that start from \(-\infty\) to -2 and from 2 to \(\infty\).

This geometric view simplifies understanding of how expressions like \(|u+v|\) function beyond just numbers, effectively translating the algebraic inequality into a visual context.