Problem 16 - Algebra 2

Question

Graph each inequality. $$ y \geq 1 $$

Step-by-Step Solution

Answer

Draw a solid line at \( y = 1 \) and shade above it.

Step 1: Identify the Inequality Type

The given inequality is \( y \geq 1 \). This means that the value of \( y \) is equal to 1 or greater. This is a linear inequality involving just one variable \( y \), which will be represented as a horizontal line on the graph.

Step 2: Draw the Boundary Line

First, draw the horizontal line \( y = 1 \) on the graph. Since the inequality is \( y \geq 1 \), this line is solid (not dashed) to indicate that points on this line satisfy the inequality.

Step 3: Shade the Solution Region

Since the inequality is \( y \geq 1 \), you need to shade the region above the line \( y = 1 \). This shaded area represents all the points where \( y \) is greater than or equal to 1. Every point in this region satisfies the inequality.

Step 4: Verify the Shaded Area

To ensure accuracy, choose a test point from the shaded area, such as \((0, 2)\). Substitute \( y = 2 \) into the inequality: \( 2 \geq 1 \), which is true. This confirms that the shading is correct.

Key Concepts

Understanding Linear Inequalities

Linear inequalities, much like linear equations, involve variables raised to the power of one. However, instead of using an equal sign, inequalities use signs such as \( > \), \( < \), \( \geq \), or \( \leq \). These symbols express a relationship where one side of the inequality is either larger, smaller, equal to or larger than, or equal to or smaller than another side.
For example, the inequality \( y \geq 1 \) indicates that any value of \( y \) is allowed as long as it is either equal to 1 or greater than 1. This is classified as a simple linear inequality since it involves a single variable raised to the first power.
Linear inequalities can be solved in ways similar to linear equations, but the focus here is not just on finding exact solutions but also understanding regions of values that satisfy the inequality.

Inequality Representation

When graphing inequalities, representation is key to understanding which values satisfy the inequality. For the inequality \( y \geq 1 \), the boundary line \( y = 1 \) is drawn on the graph. This boundary serves as a crucial visual aid because it precisely delineates the set of solutions that satisfy the inequality.
There are two types of boundary lines:

• Solid lines, which mean the boundary itself satisfies the inequality, as is the case with \( y \geq 1 \). The points on the line are part of the solution set since "equal to" is included in the inequality.
• Dashed lines, which suggest that the boundary itself is not included in the solution set, used when you encounter strict inequalities like \( y > 1 \).

By correctly representing the inequality through these boundary lines, interpreting or drawing a graph becomes simpler and more accurate.

Graphing Techniques for Inequalities

One of the most effective ways to understand and solve inequalities is by graphing them, as this visual approach makes relationships clearer. Here’s a simple step-by-step approach to graphing a linear inequality:

• **Draw the Boundary Line:** For the inequality \( y \geq 1 \), draw the horizontal line \( y = 1 \). This will be a solid line to indicate that the line is part of the solution.
• **Shade the Appropriate Region:** Since the inequality is \( y \geq 1 \), shade the entire region above the line \( y = 1 \), which includes the line itself. This area represents all the values of \( y \) that satisfy the inequality.
• **Verify the Shading:** Select a test point from the shaded region, such as \((0, 2)\). Substitute into the inequality to confirm that this point satisfies it. If true, the shading is correct. For example, \( 2 \geq 1 \), which is true.

These steps ensure that you accurately represent the set of values that solve the inequality through graphing. Mastering these techniques will make tackling inequalities much easier when graphing.