Problem 18 - Algebra A Combined Function

Question

Solve each inequality. Graph the solution set. Write each answer using solution set notation. $$ 3-7 x \geq 10-8 x $$

Step-by-Step Solution

Answer

Solution: \( \{ x \,|\, x \geq 7 \} \), Graph: closed dot at 7, shade right.

Step 1: Simplify the Inequality

We start with the inequality \( 3 - 7x \geq 10 - 8x \). First, we want to combine like terms involving \( x \) on one side. Add \( 8x \) to both sides to eliminate the \( -8x \) from the right hand side.\[ 3 - 7x + 8x \geq 10 - 8x + 8x \]Thus, the inequality becomes:\[ 3 + x \geq 10 \]

Step 2: Isolate the Variable

Now, isolate \( x \) by subtracting 3 from both sides:\[ 3 + x - 3 \geq 10 - 3 \]This simplifies the inequality to:\[ x \geq 7 \]

Step 3: Write the Solution in Set Notation

Since we now have \( x \geq 7 \), we can write the solution set in set notation as:\[ \{ x \,|\, x \geq 7 \} \]

Step 4: Graph the Solution on a Number Line

To graph \( x \geq 7 \), draw a number line and place a closed dot (because \( \geq \) means including 7) at 7 and shade everything to the right of 7 indicating all values greater than or equal to 7.

Key Concepts

Solution Set Notation

When solving inequalities, being able to express the solutions correctly is key. Solution set notation helps us make clear which values satisfy the inequality. In our example, after isolating the variable, we found that \( x \geq 7 \).

To write this in solution set notation, we use a format that specifies all elements that satisfy the condition. Generally, it appears as a set inside curly braces. Here, our solution can be conveyed as:

• \( \{ x \mid x \geq 7 \} \)

This simply means "the set of all \( x \) such that \( x \) is greater than or equal to 7." This notation is important when you want others to understand which numbers solve the inequality.

Always make sure that solution set notation reflects the conditions of the inequality accurately, so there's no confusion about which numbers are involved.

Variable Isolation

Understanding how to isolate the variable is crucial in solving inequalities and equations. Isolating the variable essentially means getting the variable alone on one side of the inequality or equation. This helps uncover which values the variable can take.

In the given problem, we began with the inequality \( 3 - 7x \geq 10 - 8x \). Our goal was to isolate \( x \).

Here's how it's done:

• Add \( 8x \) to both sides to cancel out the \( -8x \):
• \( 3 - 7x + 8x \geq 10 - 8x + 8x \)
• This simplifies to \( 3 + x \geq 10 \).

Next, to isolate \( x \), we subtract 3 from both sides:

• \( 3 + x - 3 \geq 10 - 3 \)
• This simplifies the inequality to \( x \geq 7 \).

Always remember that whatever you do to one side of the inequality, you must also do to the other side to maintain balance. This ensures that you maintain the integrity of the inequality's solution.

Graphing Inequalities

Graphing inequalities on a number line helps visualize the solution. Let's look at how to do this for the inequality \( x \geq 7 \).

Begin by drawing a simple number line with appropriate markings. To represent \( x \geq 7 \), follow these steps:

• Locate the number 7 on the line.
• Place a closed dot at 7. The dot is closed because \( x \geq 7 \) includes the number 7 itself.
• Shade the line to the right of 7 entirely, indicating all numbers greater than or equal to 7.

This graph visually demonstrates that any number 7 or higher is a solution to the inequality. Remember:

• A closed dot shows that the number is included (\( \geq \) or \( \leq \)).
• An open dot indicates exclusion (\( > \) or \( < \)).

Graphing is a straightforward way to represent solution sets and it helps ensure understanding by providing a clear visual of all possible solutions.