Problem 34 - Algebra A Combined Function

Question

Graph each inequality in two variables. $$ y \leq 1 $$

Step-by-Step Solution

Answer

Graph a solid horizontal line at \( y = 1 \) and shade below.

Step 1: Understand the Inequality

The inequality given is \( y \leq 1 \). This means that the value of \( y \) is less than or equal to 1 for any value of \( x \). The solution set includes all points on the plane where \( y \) is 1 or less.

Step 2: Graph the Boundary Line

First, we need to graph the boundary line. For \( y \leq 1 \), the boundary line is \( y = 1 \). This is a horizontal line because the value of \( y \) does not depend on \( x \). Draw a solid horizontal line across the y-axis at \( y = 1 \) to indicate that points on this line are included in the solution set.

Step 3: Shade the Solution Region

Since \( y \leq 1 \), we need to shade the region below the line \( y = 1 \). This includes all points where \( y \) is less than (but not greater than) 1. This shaded region represents all solutions to the inequality.

Key Concepts

Boundary Line

When graphing an inequality in two variables, the first step is to identify the boundary line. The boundary line helps you know exactly where the inequality switches from true to false. Think of it like a border that divides the areas where the inequality holds from those where it does not.

In the example of the inequality \( y \leq 1 \), the boundary line is \( y = 1 \). This is a straightforward horizontal line because its value does not change regardless of what \( x \) is. In other words, no matter how far left or right the line goes, \( y \) stays at 1.

By drawing this line as solid rather than dashed, we show that \( y = 1 \) itself is part of the solution. Solid lines are used for \( \leq \) and \( \geq \) symbols, indicating that points directly on the line satisfy the inequality.

Solution Region

Once the boundary line is drawn, the next step is to identify the solution region. This is the part of the graph where the inequality holds true. It's like a map showing all the areas where the inequality "lives."

For the inequality \( y \leq 1 \), the solution region includes all the points where \( y \) values are 1 or less. Since the boundary line \( y = 1 \) is solid, it is included in this region. To graph this, shade the area below the line on your graph.

Remember:

• If the inequality symbol is \(<\) or \(>\), shade above or below the boundary line respectively.
• If the inequality symbol includes equal to, as in \( \leq \) or \( \geq \), include the line itself by making it solid.

Shading helps to visually represent where the inequality is satisfied on the graph.

Inequality Symbols

Understanding the symbols in inequalities is crucial for properly graphing them. They tell us not only how the boundary line should appear but also which part of the graph to shade.

Here's a quick guide to inequality symbols:

• \( < \): Less than.
• \( \leq \): Less than or equal to.
• \( > \): Greater than.
• \( \geq \): Greater than or equal to.

For example, in the inequality \( y \leq 1 \), the "\( \leq \)" symbol means "less than or equal to," so any \( y \) value of 1 or less is included.

When graphing:

• Use a solid line for "\( \leq \)" or "\( \geq \)" to show the line is included in the solution.
• Use a dashed line for "\(<\)" or "\(>\)" when the line itself is not part of the solution.

Knowing these symbols helps make the graph accurately reflect all solutions to the inequality.