Question

The cruiser weight division in boxing is centered at 183 pounds. A boxer's weight can be as much as 7 pounds more than or less than 183 pounds. Write an absolute-value inequality for this weight requirement.

Short answer

The absolute-value inequality to represent the weight requirement for the cruiser weight division is |x - 183| ≤ 7.

Step-by-step solution

Understand what absolute-value inequality represents

An absolute-value inequality looks like this: |x - a| ≤ b. This is read as 'the distance between x and a is less than or equal to b'. In the context of this exercise, x is the weight of the boxer, a is the central weight of the division (183 pounds), and b is the maximum allowed deviation from the central weight (7 pounds).

Set up the absolute-value inequality

Based on the understanding from step 1, the absolute-value inequality for this situation is |x - 183| ≤ 7.

Interpret the Inequality

The absolute-value inequality |x - 183| ≤ 7 means that the weight of the boxer (x) must be within 7 pounds of the cruiser weight center (183 pounds). This encompasses a range from 176 pounds to 190 pounds.

Key concepts

Algebraic Inequalities

Algebraic inequalities are mathematical expressions involving symbols to indicate that one value is larger or smaller than another. They form an integral part of solving various types of real-life problems. Inequalities are denoted by signs like < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to).

The purpose of the inequality is not just to find one solution, but to determine the set of all possible solutions that satisfy the condition. In the case of weight requirements in boxing, algebraic inequalities are used to find the range of acceptable weights for boxers in a specific weight class.

The key to understanding how to set up an inequality is to identify the required conditions and translate them into a mathematical representation. For instance, if a boxer’s weight can fluctuate within a certain number of pounds from the central weight, this range can be expressed mathematically using inequalities.

By mastering algebraic inequalities, students can approach real-world scenarios methodically, think critically about the conditions given, and solve problems by finding suitable numerical ranges.

Absolute Value

The absolute value of a number is a measure of its magnitude without regard to its sign. In simple terms, it's the distance of a number from zero on the number line. Represented by two vertical lines around the number, for example, |x|, the absolute value of x is always positive or zero.

When applied to real-world contexts, like the rules for a boxing weight class, absolute values can describe allowable variations from a standard. For example, if a cruiserweight boxer must be within 7 pounds of the ideal weight of 183 pounds, this can be represented as |x - 183|. This expression essentially erases the plus or minus sign, reflecting the fact that the boxer can be either above or below the ideal weight.

Interpreting Absolute Values in Inequalities

When an absolute value is included in an inequality, it indicates that both the positive and negative variations must be considered. So, |x - 183| ≤ 7 means that x (the boxer's weight) can fall within a range 7 units above or below 183 pounds, ensuring that the concept is inclusive of all scenarios within the set bounds.

Weight Requirements in Boxing

Boxing has strict categories for weight divisions to ensure fair play and safety among competitors. Each weight class has clearly defined limits on the minimum and maximum allowable weight of the boxers. These restrictions are often described using inequalities due to the variability of human body weight.

In our exercise, the cruiserweight division sets a central weight of 183 pounds, and boxers are allowed to be within 7 pounds of this target. This margin accommodates natural fluctuations in weight while maintaining competitive balance.

Real-life Application of Absolute-Value Inequalities

Writing absolute-value inequalities to describe weight requirements ensures both ends of the weight spectrum are acknowledged. By stating |x - 183| ≤ 7, we cover boxers who might weigh slightly more or less than the central figure without having to write two separate inequalities. This notation is crucial for setting transparent, enforceable standards in sports and underlines the practical application of mathematical concepts in everyday life.