Problem 10 - Precalculus with Limits

Question

In Exercises 5-18, sketch the graph of the inequality. $$10 \geq y$$

Step-by-Step Solution

Answer

The graph of the inequality \(10 \geq y\) is a solid horizontal line at \(y = 10\), with the area below the line shaded to represent the solution region.

Step 1: Sketch the base line

Start by drawing a horizontal line at \(y = 10\). This line will be solid because the inequality includes 'equal to'.

Step 2: Identify the solution area

Since \(y\) is required to be less than or equal to 10, the area that represents the solution to the inequality will be below the line. This is the area where the \(y\)-values are less than 10.

Step 3: Shade the solution area

Shade the area below the line to represent all the possible solution points of the inequality.

Key Concepts

Sketching Inequalities

When you come across an inequality in mathematics, it often needs to be represented visually to aid understanding. Sketching inequalities on a coordinate plane involves several clear steps. To begin with, it's crucial to establish a base line, which is typically a solid line if the inequality is inclusive (such as \(y \leq 10\)) or a dashed line if not (like \(y < 10\)). This base line acts as a border that marks one of the edges of the solution area.

Once the line is in place, the next task is to identify where the solutions lie in relation to this boundary. For example, an inequality that requires \(y\) to be less than or equal to a value, such as \(10 \geq y\), will have the solution area below the line on the graph. This is because any point lying below this line will have a \(y\)-value which satisfies the inequality. To ensure clarity, it is common to shade in this solution area, indicating all possible points that are part of the solution set. Shading makes it immediately obvious to anyone reviewing the graph which side of the line contains values that satisfy the inequality.

Inequality Solutions

Inequality solutions can initially seem slightly perplexing because, unlike equations, they do not consist of a single solution. Instead, they involve a range or a set of values that satisfy the inequality's conditions. When graphing an inequality solution on the coordinate plane, the shaded area portrays this range of possibilities.

For instance, the inequality \(10 \geq y\) includes every point where the \(y\)-value is 10 or less. It does not just include \(y=10\) but also \(y=0\), \(y=-5\), and so on, down to infinitely smaller values. Ensuring that students understand this concept of a continuous range of solutions, rather than discrete points, is fundamental in interpreting and sketching inequalities. The shading on the graph represents all these possible values, thereby providing a visual representation of the solutions that is easily interpreted.

Horizontal Lines in Coordinate Plane

Understanding horizontal lines is critical when it comes to graphing inequalities or equations. A horizontal line in the coordinate plane has the same \(y\)-value across its entire length. This means that no matter what the \(x\)-value may be – whether it is 1, 100, or even -50 – the \(y\)-value will remain constant.

From a graphical perspective, these lines run left to right and are parallel to the \(x\)-axis. When you are dealing with an inequality such as \(10 \geq y\), the horizontal line drawn at \(y=10\) creates a visual reference point which represents all points where \(y\) equals 10. This line lets us see that any \(y\)-value below it – that is, any point in the shaded area – satisfies the inequality being graphed. It's a simple yet effective way of delineating between values that are part of the solution set and those that are not.