Problem 56 - 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. \(-x>-3\)

Step-by-Step Solution

Answer

The solution to the inequality \(-x>-3\) is \(x<3\). It is graphed as all points to the left of 3 (not including 3) on a number line. The solution set is \(-\infty, 3)\.

Step 1: Isolate the variable

First, one would need to isolate the variable in the inequality. For \(-x>-3\), multiply both sides of the inequality by -1. Remember, when multiplying or dividing an inequality by a negative number, the inequality sign flips. So, multiplying by -1, we get \(x<3\).

Step 2: Graph the solution

Next, plot this solution on a number line. Put a circle at 3 on the number line to represent the 'less than' condition, meaning that 3 is not included in the solution. Draw a line going to the left from 3 to represent all numbers less than 3. These are the solutions of the inequality.

Key Concepts

Multiplication Property of Inequality

Understanding the multiplication property of inequality is a key aspect of solving inequalities. In its simplest form, this principle states that when you multiply or divide both sides of an inequality by a positive number, the direction of the inequality doesn't change. For example, if you have the inequality \(2x > 6\), and you divide both sides by 2, the inequality remains in the same direction: \(x > 3\).

However, when you multiply or divide both sides by a negative number, the direction of the inequality must be reversed to maintain the truthful comparison. For instance, in the exercise \(-x > -3\), to isolate the variable, you multiply both sides by -1, resulting in the inequality flipping to \(x < 3\). This is a crucial step to prevent incorrect solutions in inequalities involving negative coefficients.

Inequality Graphing

Visualizing an inequality can vastly improve your understanding of the possible solutions. When graphing inequalities on a number line, you're illustrating the range of values that solve the inequality.

Choosing the Right Symbols

You should use an open circle to indicate that a number is not included in the solution set (for 'less than' or 'greater than'), and a closed circle for 'less than or equal to' or 'greater than or equal to' scenarios, meaning the number itself is a valid solution.

It's also important to shade the correct side of the circle to represent all numbers that are part of the solution. For example, if the inequality is \(x < 3\), an open circle at 3 and shading to the left indicates that all values less than 3 are solutions. This visual aid supplements the numerical solution with a clear graphical representation.

Number Line

A number line is an essential tool in mathematics, providing a visual representation of numbers in a linear and ordered fashion. When dealing with inequalities, number lines allow us to display the range of solutions in a way that can be quickly and easily understood.

Understanding Number Line Markings

Upon graphing, numbers less than a given value are located to the left, while numbers greater than that value are on the right. For our example \(x < 3\), every point to the left of 3 is a solution and is usually marked with a shaded line or arrow. This gives a clear indication that all numbers less than 3 satisfy the inequality. Utilizing a number line helps reinforce the concept of order and magnitude which is fundamental in understanding and solving inequalities.

Variable Isolation

When solving inequalities or equations, isolating the variable—getting the variable by itself on one side of the inequality or equation—is a top priority. This allows you to clearly identify the solution or set of solutions.

Steps for Isolation

The process typically involves inverse operations such as adding or subtracting terms from both sides, and multiplying or dividing both sides by a number (taking care of the multiplication property of inequality if dividing by a negative). In the given example, \(-x > -3\), multiplying both sides by -1 isolates the variable, giving you \(x < 3\).

Variable isolation is not only a technique but a strategic move to unveil the range or the specific number that is the answer to the problem. Whether dealing with simple or complex inequalities, mastering variable isolation is indispensable in the journey to solving mathematical puzzles.