Problem 37 - Intermediate Algebra

Question

Graph each inequality. $$ y-4.5<0 $$

Step-by-Step Solution

Answer

Graph a dashed horizontal line at \( y = 4.5 \) and shade below it.

Step 1: Understand the Inequality

The given inequality is \( y - 4.5 < 0 \). To solve for \( y \), we need to isolate \( y \) on one side of the inequality.

Step 2: Solve the Inequality

Add 4.5 to both sides of the inequality to isolate \( y \): \( y - 4.5 + 4.5 < 0 + 4.5 \)Simplifying, we get: \( y < 4.5 \)

Step 3: Graph the Solution

On a two-dimensional coordinate plane, draw a horizontal line at \( y = 4.5 \). Since the inequality is \( y < 4.5 \), use a dashed line to represent that \( y = 4.5 \) is not included in the solution. Shade the region below this line to represent all the values of \( y \) that are less than 4.5.

Key Concepts

Understanding the Coordinate Plane

The coordinate plane is an essential tool for graphing equations and inequalities. It is a two-dimensional surface formed by two number lines intersecting perpendicularly. These lines are called the x-axis (horizontal) and the y-axis (vertical). Together, they divide the plane into four quadrants.
The x-axis and y-axis meet at the origin, which is the point (0, 0). As you move to the right from the origin, the values on the x-axis become positive. Moving to the left makes them negative. Similarly, as you move up from the origin, the values on the y-axis become positive, and moving down makes them negative.
When graphing inequalities like the one in our exercise, understanding the coordinate plane helps you plot points and lines precisely. The inequalities often require you to shade a region of the plane, representing all the possible solutions.

The Process of Isolation of Variable

Isolation of a variable is a fundamental step when solving equations and inequalities. It means making the variable the subject of the equation or inequality. This helps in solving for the variable and graphing the solution.
For the given inequality, \( y - 4.5 < 0 \), the goal is to solve for \( y \). To isolate \( y \), we need to remove any numbers attached to it. In this case, subtracting 4.5 is balanced out by adding 4.5 to both sides:

• Original inequality: \( y - 4.5 < 0 \)
• Add 4.5: \( y - 4.5 + 4.5 < 0 + 4.5 \)
• Resulting in: \( y < 4.5 \)

This isolation step ensures clarity in your solution, making it easier to graph or analyze the inequality.

Graphing the Inequality Solution

After isolating the variable in an inequality, the next step is to graph the solution. This visual representation on the coordinate plane provides a clear understanding of all the possible solutions to the inequality.
For the inequality \( y < 4.5 \), start by drawing a horizontal line at \( y = 4.5 \). Since the inequality is 'less than' and not 'less than or equal to', use a dashed line. This indicates that \( y = 4.5 \) itself is not included in the solution. Then, shade the region below this line, representing all values of \( y \) that are less than 4.5.
Using a dashed line and shading appropriately are vital in showing the complete solution to the inequality visually. This graphical method supports understanding and effectively communicates the solution set.