Problem 45 - 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. \(3 x \geq-21\)

Step-by-Step Solution

Answer

The solution to the inequality \(3x \geq -21\) is \(x \geq -7\). This is represented on the number line by a coloured in dot at -7 and a line extending to the right of -7.

Step 1: Solve the inequality

In order to solve the inequality \(3x \geq -21\), we need to isolate x. We can do this by dividing both sides of the inequality by 3 because we know that any number divided by itself equals 1. So, \(3x ÷ 3 \geq -21 ÷ 3\). Simplifying this gives us \(x \geq -7\).

Step 2: Graph the solution set on the number line

To graph \(x \geq -7\) on a number line, a line is drawn and numbers are marked off. We then colour in a dot at -7, as the inequality includes the number -7, and draw a line extending to the right of -7 to represent all of the numbers greater than -7.

Key Concepts

Multiplication Property of Inequality

The multiplication property of inequality is a fundamental concept in algebra. It allows you to perform operations on inequalities while maintaining their truth.

• When you multiply or divide both sides of an inequality by a positive number, the inequality sign stays the same. This is because multiplying or dividing by a positive number keeps the order of the numbers intact.
• However, if you multiply or divide by a negative number, the inequality sign flips direction. This is because multiplying or dividing reverses the order of numbers on the number line.

In the example given, the inequality is \(3x \geq -21\). To isolate \(x\), we divide both sides by \(3\), which is a positive number. Therefore, the inequality sign does not change, and we have \(x \geq -7\). This principle is crucial for solving inequalities correctly, as it ensures you get the correct solution set.
Remember: Always pay attention to the sign of the number you multiply or divide by in inequality equations. It dictates whether you keep or flip the inequality.

Graphing Inequalities

Graphing inequalities is a way to visually represent all possible solutions of an inequality on a number line. It's a handy method to see which numbers satisfy the inequality.

• Firstly, draw a horizontal line that will serve as your number line.
• Next, you identify the solution boundary, which in this exercise is \(-7\).
• If the inequality is \( \geq \) or \( \leq \), like \(x \geq -7\), you include the number itself as part of the solution. This is shown by shading or coloring the point.
• Extend an arrow from the boundary point to represent all the numbers that satisfy the inequality condition.

For \(x \geq -7\), you color in the dot at \(-7\) since \(-7\) is part of the solution. Then, draw an arrow to the right to show that any number greater than \(-7\) is also a solution. This graphical representation helps to visually confirm the solutions are correctly spanning all numerical values that make the inequality true.

Number Line Representation

A number line is a simple and effective way to display the range of solutions for inequalities. Its visual simplicity makes it easier to understand the solutions.

• To start, draw a straight, horizontal line and label it with evenly spaced numbers. Choose marks covering the range of the inequality’s solution region.
• Identify the critical point. For \(x \geq -7\), \(-7\) is our starting point. Place a filled dot on \(-7\) because the solution includes this value.
• Next, extend a line or arrow starting from \(-7\), moving towards the direction of the solutions. In this case, the arrow goes to the right, indicating values greater than \(-7\).

This visualization not only aids in understanding the specific range of solutions but also gives clarity to the concept of inequality. A number line effectively depicts why certain numbers are solutions, by showing their position relative to the critical point, like \(-7\) here.