Problem 11 - Elementary Algebra

Question

Solve the inequalities by graphing. $$ y \leq 4 $$

Step-by-Step Solution

Answer

Question: Identify the solution region for the inequality \(y \leq 4\) on the coordinate plane. Answer: The solution region for the inequality \(y \leq 4\) is the region below the horizontal line y = 4, including the line itself.

Step 1: Setup the coordinate plane

First, we need to set up a coordinate plane. Draw the x-axis (horizontal) and the y-axis (vertical), and label them accordingly.

Step 2: Graph the line y = 4

Now, we graph the line y = 4. This is a horizontal line that passes through all points on the plane whose y-coordinate is 4. To draw the line, simply plot a few points where y = 4 (for example, (0,4), (1,4), (-1,4)) and connect them with a straight line. Since our original inequality is \(\leq\), we will use a solid line for y = 4, which indicates that the points on the line are included in the solution.

Step 3: Identify the solution region

The inequality is \(y \leq 4\). On the coordinate plane, this means that we are looking for the region where the y-values are less than or equal to 4. This region is below the horizontal line y = 4 (including the line itself).

Step 4: Shade the solution region

Shade the region below the line y = 4, including the line itself, to represent the solution region for the inequality. This indicates that all points in the shaded region will satisfy the given inequality. Now the inequality is solved by graphing, and the shaded region in the coordinate plane represents the set of solutions for the inequality \(y \leq 4\).

Key Concepts

coordinate plane

A coordinate plane is a two-dimensional surface that allows us to graph equations and inequalities visually. It consists of two perpendicular axes:

• The x-axis, which runs horizontally.
• The y-axis, which runs vertically.

These axes intersect at a point called the origin, denoted as (0,0). Each point on the plane is defined by a pair of numbers, known as coordinates. These coordinates specify the position of the point relative to the x-axis and y-axis.

When working with the coordinate plane, it's important to label the axes so that you know which direction is positive or negative. Typically, numbers increase to the right on the x-axis and upwards on the y-axis. By plotting points in this system, you can represent mathematical relationships and graph linear equations like lines.

inequalities

Inequalities help us compare two different expressions. They show whether one expression is greater than, less than, equal to, or not equal to another expression. The symbols used in inequalities include:

• \(>\): greater than
• \(<\): less than
• \(\geq\): greater than or equal to
• \(\leq\): less than or equal to

A key feature of inequalities on the coordinate plane is that they divide the plane into two regions. One region is where the inequality holds true, and the other is where it does not hold true.

When solving inequalities by graphing, if the inequality includes equality (using \(\geq\) or \(\leq\)), the boundary line is drawn as a solid line. This indicates that points on the line are solutions to the inequality. If the inequality does not include equality (using \(>\) or \(<\)), a dashed line is used instead.

solution region

The solution region of an inequality on a graph is the area that satisfies the inequality. For example, for the inequality \(y \leq 4\), the solution region includes all points on the coordinate plane where the y-coordinate is less than or equal to 4.

To find the solution region:

• First, graph the boundary line (e.g., \(y = 4\)). If the inequality is \(\leq\) or \(\geq\), use a solid line.
• Identify which side of the line represents the solutions to the inequality. For \(y \leq 4\), it's below the line since those y-values satisfy the inequality.
• Finally, shade this area to represent the solution region. Shading indicates which points fulfill the inequality. Points in this shaded region are part of the solution set.

Shading not only helps in visualization but also ensures clear communication of the solution region to anyone interpreting the graph.