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When we talk about algebraic inequalities, we are referring to expressions that show the relationship between two values where one is not necessarily equal to the other. Unlike equations that state that two expressions are equal, inequalities use symbols to suggest that one value is greater than, less than, equal to, or not equal to another value.

In the context of the New River Gorge Bridge problem, we are given a specific length, 1700 feet, and we need to express that any other steel arch bridge would be either shorter than or exactly as long as this bridge. To represent this scenario, we use the inequality notation which in this case is the '\textless=' symbol, indicating 'less than or equal to'. The algebraic inequality for the length of any other steel arch bridge would be written as: \( l \leq 1700 \).

This tells us that the value for \( l \), the length of any other steel arch bridge, should be a number that is no greater than 1700. It incorporates all possible lengths from the smallest conceivable bridge to the 1700-foot length of the New River Gorge Bridge.

Understanding inequality notation is key to solving and graphing inequalities. There are several symbols used to represent inequalities:

• '\textless' means 'less than.'
• '\textgreater' means 'greater than.'
• '\(\leq\)' means 'less than or equal to.'
• '\(\geq\)' means 'greater than or equal to.'
• '\(eq\)' means 'not equal to.'

Each of these symbols help to define the nature of the relationship between quantities or variables.

In our exercise, the notation '\( l \leq 1700 \)' succinctly conveys that the length of any other steel arch bridge is no longer than 1700 feet. Here, the '\(\leq\)' symbol is appropriate as it includes the possibility of another bridge being exactly 1700 feet long, just like the New River Gorge Bridge—the current record holder for the longest steel arch bridge.

The number line is a visual tool for representing numbers and inequalities. When graphing inequalities, the number line helps to illustrate the range of possible values that satisfy the inequality. Here's how you can graph the inequality '\( l \leq 1700 \)' representing the lengths of steel arch bridges shorter than or equal to the New River Gorge Bridge on a number line:

Start by drawing a horizontal line and marking off intervals to represent feet. You would place a point at 1700 to represent the length of the New River Gorge Bridge. Since the inequality indicates 'less than or equal to,' you would draw a closed circle at 1700, signaling that this value is included in the solution set. Then, shade the line to the left of 1700 all the way down to 0, showing that any length within this shaded area satisfies the inequality. The shaded region visually represents all possible lengths of other steel arch bridges that are no longer than 1700 feet, clearly demonstrating the solution to the inequality problem.