Question

The Verrazano-Narrows Bridge in New York City is the longest suspension bridge in North America, with a main span of 4260 feet. Write an inequality that describes the length in feet \(x\) of every other suspension bridge in North America. Graph the inequality on a number line.

Short answer

\(x \leq 4260\) is the inequality that describes the length \(x\) in feet of every other suspension bridge in North America. On a number line, it's represented by marking 4260 with a closed circle and having an arrow pointing to the left to indicate that \(x\) includes all lengths less than or equal to 4260.

Step-by-step solution

Define the Variable

Let's define \(x\) as the length in feet of any other suspension bridge in North America.

Formulate the Inequality

The problem states that the Verrazano-Narrows Bridge is the longest suspension bridge in North America with a length of 4260 feet. Therefore, every other bridge will be shorter or, in mathematical terms, their length will be less than or equal to 4260 feet. This can be represented by the inequality \(x \leq 4260\).

Graph the Inequality on a Number Line

To illustrate this inequality on a number line, draw a line and mark a point at 4260. As \(x\) can be equal to 4260, make this a closed or filled-in circle. To show \(x\) is any value less than 4260, draw an arrow pointing left from the 4260 mark, indicating all the numbers to the left. This is the graphical representation of the inequality.

Key concepts

Understanding the Suspension Bridge Problem

When mentioning the 'suspension bridge problem', we're talking about problems that involve comparing a specific example with a set of others based on a particular characteristic—in this case, the length of suspension bridges.

In our exercise, the Verrazano-Narrows Bridge serves as the point of reference. Its length is the longest among suspension bridges in North America, which establishes a 'benchmark' that other bridges can be compared to. Such problems are common in algebra, often serving to introduce students to the concept of inequalities. Here, the problem is to express mathematically the lengths of all other suspension bridges in North America with an inequality.

To make sense of this scenario, one must understand that an inequality is a mathematical way to represent how numbers compare to one another. It's not just about finding equality or the exact measure but describing a range of possible values that meet certain conditions.

Graphing Inequalities

Graphing inequalities is a vital tool in algebra that visually represents the range of possible solutions to an inequality. It's different from graphing an equation because an inequality shows a continuum of values rather than specific points.

For the example of the suspension bridge, we've determined that the length of other bridges ($x$) is less than or equal to 4260 feet. To graph this, imagine a horizontal line, which is our number line. Place a solid dot on the number representing 4260 because the inequality includes bridges that are exactly 4260 feet long. The solid dot is crucial as it indicates that this value is part of the solution set, known as a 'closed dot'.

Then, to represent all possible lengths less than this value, we draw a line or arrow extending to the left of this dot towards smaller numbers. This visual aid helps students immediately see all the possible values of $x$ that satisfy the inequality, offering a solid understanding of the solutions without solely relying on abstract numbers.

Number Line Representation

A number line is one of the simplest yet most powerful tools in mathematics to represent and understand various concepts, including inequalities.

In our suspension bridge problem, the number line serves as a means to display all lengths that are possible for other suspension bridges. It provides a quick visual representation that conveys which numbers are part of the solution set for the inequality $x$ is less than or equal to 4260. You can think of the number line as a 'timeline' where smaller numbers (to the left) represent shorter bridges and the point at 4260 represents the Verrazano-Narrows Bridge.

This number line visualization is instrumental for students to grasp how inequalities work. It makes the concept tangible by showing that the area to the left of the closed dot at 4260 includes all bridge lengths that are less than or equal to that of the Verrazano-Narrows Bridge. This helps in understanding not just this problem, but the nature of inequalities as a whole, laying the groundwork for solving more complex algebra problems in the future.