Problem 40 - Introductory Algebra for College Students

Question

Use the multiplication property of inequality to solve each inequality and graph the solution set on a number line. \(\frac{1}{2} x>3\)

Step-by-Step Solution

Answer

Solving the inequality gives \(x > 6\), indicating that all numbers greater than \(6\) are solutions to this inequality.

Step 1: Isolate the variable

To isolate the variable \(x\), multiply both sides of the inequality by the reciprocal of \(\frac{1}{2}\), which is \(2\). \[2 \times \(\frac{1}{2} x\)>2 \times 3 => x > 6\]

Step 2: Plot the inequality

On the number line, plot an open dot at \(6\), and draw an arrow to the right, representing all numbers greater than \(6\). The open dot means that \(6\) is not included in the solution set.

Key Concepts

Multiplication Property of Inequality

Understanding how to solve inequalities might seem tricky at first. A vital tool for this process is the multiplication property of inequality. This property states that when you multiply or divide both sides of an inequality by the same positive number, the direction of the inequality remains unchanged.

So, in simple terms:

• If you have \( a > b \), and you multiply both sides by a positive number \( c \), the inequality becomes \( ac > bc \).
• The inequality maintains its direction, ensuring the same relationship between the quantities.

In the given exercise, to solve \( \frac{1}{2} x > 3 \), the goal is to isolate \( x \) on one side. We do this by multiplying both sides by the reciprocal of \( \frac{1}{2} \), which is \( 2 \). After applying the multiplication, the inequality becomes \( x > 6 \).

Keep in mind, if you ever multiply or divide by a negative number, the inequality sign flips. However, in our exercise, we dealt with a positive number, hence, we keep the sign as it is.

Graphing Solution Sets

Once the inequality is solved, the next step is graphing the solution set. Graphing helps us visualize the range of possible solutions that satisfy the inequality.

For the inequality \( x > 6 \), the solution set consists of all numbers greater than \( 6 \).
Let's break down how to graph this:

• Identify the critical point: The inequality revolves around 6 (the number on the right side of the inequality).
• Use an open dot: Since \( 6 \) itself is not part of the solution (as indicated by the "greater than" symbol without an "equal to"), place an open dot on the number line at \( 6 \).
• Draw the arrow: To show that the solutions extend beyond \( 6 \), draw a line starting at the dot and pointing to the right.

This graph visually communicates that every number greater than \( 6 \) meets the condition set by the inequality. It's a handy way to demonstrate solution sets, especially for inequalities that include infinitely many numbers.

Number Line

The number line is a simple yet powerful tool for graphing inequalities. It provides a straightforward way to display which numbers are included in a solution set.

Here’s how to effectively use a number line to represent solutions:

• Order: The number line is laid out as a horizontal line, with numbers increasing to the right and decreasing to the left.
• Open vs. Closed Dots: An open dot indicates a number not included in the solution, while a closed dot indicates an included number.
• Arrow indicating direction: To represent ranges of solutions, arrows extend from certain numbers to show the direction and scope of possible solutions.

In our exercise, because the inequality is \( x > 6 \), we use an open dot at \( 6 \) to show it's not included. Then, the arrow going to the right highlights that solutions include all numbers greater than \( 6 \).

Number lines offer a visual advantage, making abstract solutions more tangible and accessible, particularly for those tackling inequalities for the first time.