Problem 15 - Algebra A Combined Function

Question

Solve each inequality. Graph the solution set. Write each answer using solution set notation. $$ -9+y<0 $$

Step-by-Step Solution

Answer

\(y < 9\); Graph: open circle at 9, shaded left; Set: \(\{ y \; | \; y < 9 \}\).

Step 1: Isolate the variable

Start with the given inequality: \ \(-9 + y < 0\). \ To isolate \(y\), we need to add 9 to both sides of the inequality. \ This gives us: \ \(y < 9\).

Step 2: Graph the solution set

Now we need to graph the solution. On a number line, draw an open circle at 9, indicating that 9 is not included in the solution set. Shade the region to the left of 9, representing all numbers less than 9.

Step 3: Write in solution set notation

The solution in solution set notation is written as: \ \(\{ y \; | \; y < 9 \}\). \ This means we are including all values of \(y\) that are less than 9.

Key Concepts

Understanding Solution Set Notation

When we solve inequalities, it's important to express our answers clearly and efficiently. This is where solution set notation comes in. Solution set notation tells us all the possible values that satisfy the inequality. In our example, the inequality was \(-9 + y < 0\). After solving, we found that \(y < 9\).
In solution set notation, this is written as:

• \(\{ y \; | \; y < 9 \}\)

This notation means "the set of all \(y\) such that \(y\) is less than 9."
It's a compact and formal way to represent the solution, making it a handy tool in mathematics to convey a range of values that a variable can take.

Mastering the Isolation of Variables

One of the most essential skills in solving inequalities is isolating variables. This means getting the variable by itself on one side of the inequality symbol. Let's look at our example: \(-9 + y < 0\).
The goal is to isolate \(y\). Here's how:

• Add 9 to both sides to remove \(-9\) from the left.
• This gives: \(y < 9\).

By performing the same operation on both sides, we maintain the balance of the inequality.
Isolating the variable helps us to understand which values make the inequality true, setting the stage for further analysis and graphing.

Graphing Inequalities Made Simple

Graphing the solution of an inequality is a visual way to represent its solutions. After isolating the variable, we had \(y < 9\).
Here's how to graph it on a number line:

• Draw a number line.
• Put an open circle on 9. The open circle signifies that 9 is not included in the solution.
• Shade the region to the left of 9. This shading indicates all numbers less than 9.

Visualizing inequalities on a graph helps in comprehending the range of solutions more intuitively.
It also assists in checking the solutions quickly, especially when dealing with compound inequalities or constraints.