Problem 44 - Introductory and Intermediate Algebra for College Students 4th

Question

Use the multiplication property of inequality to solve each inequality and graph the solution set on a number line. $$6 x<18$$

Step-by-Step Solution

Answer

The solution is \(x < 3\). In the graph, an open circle is placed at 3 and line is drawn to the left, indicating the solutions.

Step 1: Isolate variable \(x\)

To isolate \(x\), divide both sides of the inequality by 6. This gives: \[x < \frac{18}{6}\].

Step 2: Simplify and Solve the inequality

Simplify \(\frac{18}{6}\) to get 3. So the solution of inequality is \(x < 3\).

Step 3: Graph the solution on a number line

Draw a number line marking the number 3. Place an open circle at 3 (as \(x\) is strictly less than 3, and does not include 3) and shade or line to the left to show all the possible values of \(x\).

Key Concepts

Multiplication Property of Inequality

The multiplication property of inequality is a useful tool that helps us manipulate inequalities to find solutions. This property states that if you multiply or divide both sides of an inequality by the same positive number, the direction of the inequality remains the same. However, if you multiply or divide by a negative number, the inequality sign must be flipped.
For example, if we have the inequality \(6x < 18\), we can solve for \(x\) by dividing both sides by 6 (a positive number), resulting in \(x < 3\). Here, the inequality sign stays as '<' because we divided by a positive number.
Understanding this property is crucial for solving inequalities correctly, especially when negative numbers are involved, as you will need to reverse the inequality sign. Always keep this rule in mind, so you won't make any mistakes in your calculations.

Graphing Inequalities

Graphing inequalities is a visual method to show all possible solutions of an inequality on a number line. When you graph inequalities, you're essentially displaying a range of numbers that satisfy the condition given by the inequality.
To graph the solution of \(x < 3\), we start by drawing a basic number line and marking the critical point, which is 3 in this case. Since the inequality is strict (it doesn't include 3 itself), we represent this by placing an open circle at 3. This indicates that 3 is not part of the solution set.
After marking your critical point, you need to shade the region on the number line that represents the solution set. For \(x < 3\), you would shade the line to the left of 3, showing all numbers less than 3 are part of the solution.
By following these steps, you visualize the inequality, making it easier to understand the range of values that satisfy it.

Number Line

A number line is a straight horizontal line that is used in mathematics to represent numbers as points spaced equally along its length. It is a handy tool for visualizing mathematical concepts, particularly when working with inequalities.
On a number line, numbers increase as you move to the right and decrease when moving to the left. This linear setup helps in graphically representing inequalities and understanding the relationship between different numbers.
For representing inequalities like \(x < 3\), the number line serves as a simple yet effective way to show where the solution lies. You use open circles or closed circles to denote whether endpoints are included in the solution set. For an inequality such as \(x < 3\), an open circle is drawn at 3 to signify it is not included, while shading to the left represents all values smaller than 3.
With practice, using a number line becomes intuitive and makes solving and graphing inequalities straightforward and accessible.