Problem 25 - Introductory and Intermediate Algebra for College Students 4th

Question

Use the addition property of inequality to solve each inequality and graph the solution set on a number line. $$y-2<0$$

Step-by-Step Solution

Answer

The solution to the inequality \(y-2<0\) is \(y<2\). This means, any number less than 2 is a solution to the given inequality.

Step 1: Isolate the Variable

To isolate the variable \(y\), add 2 on both sides of inequality. This step is based on the addition property of inequality, which states that the inequality holds true when the same amount is added to both sides.So, the inequality becomes:\(y-2+2 < 0+2\)Simplify to get: \(y < 2\)

Step 2: Represent the Solution on a Number Line

Now the solution set is any number less than 2. To represent it on a number line, draw a line, indicating the number 2 with a small circle (as 2 is not included in the solution), and draw an arrow pointing to the left from 2 indicating all numbers less than 2 are solutions for this inequality.

Key Concepts

Addition Property of Inequality

The addition property of inequality is a simple but powerful tool in solving inequalities. To understand it better, think of inequalities as balance scales. If both sides of an inequality are like two objects on a scale, adding the same weight to each keeps the balance unchanged.

Here's how it works:

• If you have an inequality like \(y - 2 < 0\), you can add the same number, say 2, to both sides without flipping the inequality sign.
• This results in \(y < 2\), which is easier to understand.

The goal is to isolate the variable (in this case, \(y\)) on one side of the inequality to see its potential values. Performing the operation equally on both sides ensures that the inequality's integrity is maintained. Understanding and applying this property is essential for working with inequalities successfully.

Graphing Inequalities

Once you've simplified the inequality, it's beneficial to graph it to visualize the solution set. Graphing inequalities helps in understanding which numbers satisfy the condition.Let’s explore how to graph \(y < 2\):

• Identify the critical number, which is 2 in this case.
• Since \(y < 2\), and not \(y \leq 2\), the number 2 itself is not a part of the solution. You'll represent this by drawing a circle around 2.
• Next, you'll draw an arrow from the circle that extends to the left, indicating all numbers smaller than 2 satisfy the inequality.

This visual helps in grasping the concept of the inequality quickly and reinforces what you've solved algebraically.

Number Line Representation

Representing solutions on a number line is a visual technique to show all values that satisfy an inequality.Here's how to create a number line representation for \(y < 2\):

• Start by drawing a horizontal line to represent your number line.
• Place a mark at 2, the boundary number, and draw a small open circle on it. This circle signifies \(2\) itself is not included in the solution set.
• Next, draw a straight line or arrow to the left from the open circle. This arrow indicates all the numbers less than 2 are included.

This method is not only useful for inequalities but also for depicting solutions in real-world problems, helping students to see and understand the range of possible solutions intuitively.