Problem 26 - Introductory Algebra for College Students

Question

Use the addition property of inequality to solve each inequality and graph the solution set on a number line. \(y+3 \geq 0\)

Step-by-Step Solution

Answer

The solution to the inequality is \(y \geq -3\), represented by a closed circle at -3 and shading to the right on the number line.

Step 1: Isolate the variable 'y'

To isolate 'y', subtract 3 from both sides of the inequality. This gives us: \(y + 3 - 3 \geq 0 - 3\), which simplifies to \(y \geq -3\)

Step 2: Graph the solution on a number line

We need to draw a number line and mark the point -3 as our boundary. Because our solution includes 'equal to', we put a closed circle on -3. The inequality \(y \geq -3\) means that 'y' is greater than or equal to -3, so we shade the number line to the right of -3, showing that all these numbers are solutions to the inequality.

Key Concepts

Addition Property of Inequality

The addition property of inequality is a fundamental rule used in mathematics to maintain the balance of an inequality. It states that adding or subtracting the same number from both sides of an inequality does not change the inequality's direction. For example, if you have an inequality expressed as \(a > b\), you can add the same number, say \(c\), to both sides, resulting in \(a + c > b + c\). This property ensures the inequality stays true and is crucial for solving inequalities.

• It helps maintain the structure and balance of the inequality.
• It allows you to isolate the variable on one side of the inequality.

Understanding how to use the addition property is decisive for manipulating inequalities and finding the value range for the variable involved.

Number Line Graphing

Number line graphing is a visual technique used to show solutions to inequalities. It is powerful because it gives a clear representation of a solution set. To graph an inequality like \(y \geq -3\), follow these steps:

• Draw a horizontal line, your number line, with numbers incrementally labeled along the axis.
• Identify the critical point from your inequality; in this case, it's -3.
• Since the inequality includes "greater than or equal to," you draw a closed circle at -3.

Once the critical point is marked, shade or highlight the line to the right of -3. This shading represents all the possible solutions for the inequality \(y \geq -3\). Number line graphing is an effective tool to visualize and verify the solution set.

Variable Isolation

Variable isolation is a key step in solving inequalities effectively. The goal is to get the variable by itself on one side of the inequality symbol. This involves performing operations such as addition, subtraction, multiplication, or division on both sides of the inequality.

• Always perform the same operation on both sides to maintain equality.
• In the inequality \(y + 3 \geq 0\), subtract 3 from both sides to isolate \(y\). This gives \(y \geq -3\).

Isolating the variable helps simplify the inequality into a form that is easy to interpret and graph. Understanding how to isolate variables is essential, as it paves the way for solving more complex inequalities and equations.