Problem 39 - 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. \(\frac{1}{2} x<4\)

Step-by-Step Solution

Answer

The solution to the inequality is \(x < 8\). In the number line representation, the arrow points towards the left side of 8, indicating all values less than 8 are valid solutions to the inequality.

Step 1: Isolate the Variable

Start by isolating \(x\) on one side of the inequality. In the given inequality \(\frac{1}{2} x < 4\), to isolate \(x\), multiply both sides by 2(the reciprocal of \(\frac{1}{2}\)). This gives: \[2 * \frac{1}{2}x < 2 * 4\] which simplifies to \[x < 8\]

Step 2: Represent the Solution on Number Line

After obtaining the solution \(x < 8\), represent it on the number line so as visualize the solution set. On the number line, indicate 8 with a small circle. Since \(x\) is less than 8, the shading (or arrow) should be towards the left side of 8, representing all values less than 8 are the solution.

Key Concepts

Multiplication Property of Inequality

Understanding the multiplication property of inequality is crucial when solving inequalities. This property states that you can multiply or divide both sides of an inequality by the same positive number without changing the inequality's direction. For instance, if you have \( a < b \), and you multiply both sides by a positive number \( c \), the inequality remains true: \( ac < bc \). However, it's important to note that if you multiply or divide by a negative number, the direction of the inequality flips.

In the exercise \( \frac{1}{2} x < 4 \) you apply this property by multiplying both sides by 2, the reciprocal of \( \frac{1}{2}\), ensuring that the variable \(x\) is isolated without affecting the truth of the original inequality. The multiplication process maintains the inequality direction because 2 is positive. If we were to multiply by a negative, we'd need to reverse the inequality sign to maintain the correct relationship.

Isolating the Variable

The goal of isolating the variable when solving any equation or inequality is to have the variable on one side and the numbers on the other. This step is fundamental because it simplifies the problem and gives you a clear solution. In the inequality \( \frac{1}{2} x < 4 \), we want to find out for which values of \( x \) the inequality holds true.

To isolate \( x \) in this particular inequality, you multiply both sides by 2, enabling the fractions to cancel out on the left side, leaving you with \( x \) alone. As a result, you get \( x < 8 \), which is a much simpler way to view and understand the range of solutions for \( x \). When isolating the variable, always perform the same operation on both sides of the inequality to maintain balance.

Number Line Representation

Using a number line representation helps visualize the set of solutions for an inequality. It is a graphical tool that allows you to see all the possible values that fulfill the inequality's conditions. When you have the isolated inequality, like \( x < 8 \), you can denote this on the number line.

To properly graph \( x < 8 \), you would draw a number line and put a small open circle at 8 to indicate that this number is not included in the solution set (because the inequality is strictly less than, not less than or equal to). From the 8, you would shade or draw an arrow to the left to show all the values of \( x \) that make the inequality true (all the numbers less than 8). This visual aid solidifies your understanding of the possible values for \( x \) and makes it easier to grasp the concept of inequality solutions.