Question

The world record for the largest Pacific blue marlin is 1376 pounds. It was caught in Kaaiwi Point, Kona, Hawaii. Let \(M\) represent the weight of a Pacific blue marlin that has been caught. Write an inequality for \(M .\) Graph the inequality. P Source: International Game Fish Association

Short answer

The inequality representing the weight of a Pacific blue marlin that has been caught is \(M \leq 1376\). The graph is a number line filled at 1376 and extending to the left.

Step-by-step solution

Write the Inequality

Given the world record as 1376 pounds, any caught Pacific blue marlin should weigh less than or equal to this. Therefore, an appropriate inequality is \(M \leq 1376\).

Graph the Inequality

To graph the inequality \(M \leq 1376\), a number line is needed. The number line will be drawn and 1376 will be marked on it. The point representing 1376 will be filled (indicating 'equal to') and a line will be drawn extending to the left (indicating 'less than') from this point towards the lower numbers.

Key concepts

Algebraic Inequalities

Understanding algebraic inequalities is a fundamental part of algebra. These inequalities express a relationship between two values where one is not strictly equal to the other but is either less than, greater than, less than or equal to, or greater than or equal to the other value. The symbol \<\ indicates 'less than', \>\ symbolizes 'greater than', while \( \leq \)\ and \( \geq \)\ stand for 'less than or equal to' and 'greater than or equal to' respectively.

For example, the inequality \( M \leq 1376 \)\ describes a relationship where the weight of a captured Pacific blue marlin, denoted by \( M \), should not exceed 1376 pounds. Such inequalities can be solved to find the range of possible values that satisfy the condition. Solving an inequality typically involves isolating the variable on one side and interpreting the results within a real-world context or on a number line for visual understanding.

Number Line Representation

A number line is a visual representation that helps to easily understand the range of values that satisfy an inequality. When graphing an inequality such as \( M \leq 1376 \)\, you begin by drawing a horizontal line with numbers indicating the possible weights of marlins in pounds.

Marking the Inequality on the Number Line

The point corresponding to 1376 is marked on your number line. Since the inequality includes 'equal to' (\leq), that point is colored in or includes a closed circle, indicating that 1376 is a part of the solution set. Next, a line or arrow is drawn to the left of 1376 to show all numbers less than 1376 are also in the solution set, as they also represent possible weights of blue marlins, which satisfy the inequality.

Inequalities in Real-World Context

Inequalities are not just theoretical constructs; they often represent real-world situations and constraints. The inequality \( M \leq 1376 \)\ corresponds to the understandable fact that no caught Pacific blue marlin can weigh more than the world record which is 1376 pounds.

In various contexts, such as in economics, engineering, or health sciences, inequalities are used to describe constraints like budgets, physical limitations, or nutritional recommendations. They allow for understanding the range within which a certain variable (such as weight, cost, or calorie intake) must fall. By learning to interpret and graph inequalities, students can better understand and analyze these real-life situations, helping with decision-making and problem-solving skills.