Problem 41 - Algebra 1: Concepts and Skills

Question

Graph the inequality. $$y \geq 5$$

Step-by-Step Solution

Answer

The graph is a solid horizontal line at \(y = 5\) with the region above the line filled in.

Step 1: Draw the boundary line

First, draw a horizontal line at \(y = 5\). This will be a solid line because the inequality is inclusive (\(\geq\), not just \(>\)). That means that all points on this line are part of the solution set.

Step 2: Shade the solution area

After drawing the line, shade the region above the line because we're talking about \(y\)-values that are greater than 5.

Step 3: Check a point

A validation step is to check a point in the shaded region to ensure it satisfies the inequality. Any point in the shaded area will do, like (0,6). Substituting these values into the inequality \(y \geq 5\) would produce 6 >= 5, which is true, so the point is part of the solution.

Key Concepts

Linear Inequalities

Linear inequalities such as \(y \geq 5\) are expressions that show a relationship of greater than or less than between two quantities. Unlike linear equations, which determine a line or a set of points, linear inequalities create a region on a graph. When tackling these inequalities, it's important to recognize the inequality sign, as it dictates whether you will include certain boundary points or not.
Linear inequalities can take different forms, like \(x + 2 > 5\) or \(3y \leq 6x - 4\). Each inequality represents a relation with two parts: the linear expression and the inequality sign. These two parts together tell us how to graph the inequality on the coordinate plane. It's crucial to understand whether the boundary line, on which these functions are based, is included in the solution set, which will be determined by whether you use a solid or dashed line.

Coordinate Plane

The coordinate plane is an essential part of graphing linear inequalities. This two-dimensional surface is defined by a horizontal line called the x-axis and a vertical line known as the y-axis. They intersect at a point called the origin, labeled as \( (0,0) \).
The coordinate plane is separated into four quadrants, which helps in organizing points and determining positions. When graphing an inequality, you place points based on their coordinates \((x, y)\) and apply these points to plot the boundary line. Remember that in the exercise, the line \(y = 5\) is essential for graphing \(y \geq 5\). This line is horizontal and cuts across the plane parallel to the x-axis at \(y = 5\). Understanding how to navigate and plot on the coordinate plane is crucial for tackling inequalities.

Solution Set

The solution set for a linear inequality encompasses not just the boundary line, but also the infinite number of points that satisfy the inequality condition.
In the case of \(y \geq 5\), the solution set includes all points on the line \(y = 5\), as well as every point above this line on the y-axis.

• Solution sets can be identified by substituting points into the inequality. If true, the point is part of the solution set.
• For \(y \geq 5\), example points like \((0,6)\) or even \((2,7)\) confirm being within the set because they satisfy the condition \(y \geq 5\).

When evaluating if a point is in a solution set, you only need to check one point, as consistently similar points within the shaded region will also satisfy the inequality.

Shading Regions

Shading regions is a vital step in graphing inequalities. Once the boundary line is drawn, shading helps to visualize the entire set of solutions to the inequality.
For \(y \geq 5\), you will shade the area above the line \(y = 5\).

• The shaded region represents all possible solutions that satisfy the inequality \(y \geq 5\).
• A simple way to confirm the right region is to test a point from either side of the line. If it satisfies the inequality, shade that side.

Solid lines accompany inequalities with "greater than or equal to" or "less than or equal to" because these include the boundary line itself. The shaded space visually confirms where the inequality holds true across the coordinate plane.