Problem 79 - Introductory and Intermediate Algebra for College Students 4th

Question

Use both the addition and multiplication properties of inequality to solve each inequality and graph the solution set on a number line. $$1-\frac{x}{2}>4$$

Step-by-Step Solution

Answer

The solution to the inequality \(1-\frac{x}{2}>4\) is \(x<-6\).

Step 1: Rearrange the inequality

Start by subtracting 1 from both sides of the inequality to remove the constant from the left side of the inequality. This gives us \(-\frac{x}{2}>3\).

Step 2: Isolate the x variable

Multiply both sides of the inequality by -2 to get the variable \(x\) alone. However, remember to flip the inequality sign since multiplying an inequality by a negative number reverses the direction of the inequality. Therefore, the new inequality becomes \(x<-6\).

Step 3: Graph the solution

On a number line, a open circle at -6 and a line extending to the left from -6, indicates the solution set includes all numbers greater than -6.

Key Concepts

Addition Property of Inequality

To solve inequalities, the addition property of inequality comes in handy. This property states that you can add (or subtract) the same number from both sides of an inequality without changing its direction.
In our exercise, the inequality began as \(1 - \frac{x}{2} > 4\). To simplify this, we subtract 1 from both sides, leading to the new inequality \(-\frac{x}{2} > 3\). This subtraction does not change the inequality, thanks to the addition property.

• Addition or subtraction of a value affects both sides equally.
• The inequality sign remains unchanged after such operations.
• This step helps in isolating terms containing the variable.

Applying this property effectively lays the groundwork for further manipulation, ensuring that our inequality remains balanced throughout the solution process.

Multiplication Property of Inequality

The multiplication property of inequality is crucial when dealing with inequalities, especially when isolation of a variable involves multiplication or division. According to this property, when you multiply or divide both sides of an inequality by a positive number, the inequality remains unchanged. However, if you multiply or divide by a negative number, the direction of the inequality must be reversed.
In our current problem, after applying the addition property, we reach \(-\frac{x}{2} > 3\). To isolate \(x\), we multiply both sides by -2, necessitating a flip of the inequality sign. So, \(x < -6\) becomes the inequality as a result of applying the multiplication property.

• Makes it possible to isolate variables effectively.
• Requires careful attention to switching inequality signs when using negative numbers.
• Ensures the inequality's solution set accurately reflects the conditions set by the inequality.

Understanding and using this property allows you to solve inequalities confidently.

Graphing Inequalities

Graphing inequalities is a visual way of representing solutions on a number line, helping to easily identify the range of possible values. Once you solve an inequality expression, like \(x < -6\), the next step is often to graph the solution set.
For our exercise solution, we draw a number line, locate -6, and place an open circle on it. The open circle indicates that -6 is not included in the solution set. We then shade the line segment extending leftward from -6.

• Use open circles for exclusion and closed circles for inclusion at specific points.
• Direction of shading indicates whether the solution set consists of values less than or greater than a point.
• Graphs offer immediate visual cues about the inequality's solution range.

Graphing forms an integral part of understanding and interpreting inequalities, helping convert abstract concepts into visual aids for better comprehension.