Problem 24 - Introductory Algebra for College Students

Question

Use the addition property of inequality to solve each inequality and graph the solution set on a number line. \(x-5 \geq 2\)

Step-by-Step Solution

Answer

The solution set for the inequality \(x-5 \geq 2\) is \(x \geq 7\). This solution can be represented on a number line with a closed circle on 7 and an arrow pointing towards the right.

Step 1: Applying the Addition Property of Inequality

The first step involves using the addition property of inequality to isolate \(x\). We add 5 to each side of the inequality to balance it out. Then, \(x-5+5 \geq 2+5\), which simplifies to \(x \geq 7\).

Step 2: Graphing the Solution Set

A number line is drawn with numbers spanned to accommodate all the possible solutions. A closed circle is placed on 7 with an arrow pointing towards the right. This indicates that 7 and all numbers larger than 7 are part of the solution set. The circle is filled/closed as 7 is included in the solution set (indicated by the 'greater than or equal to' inequality symbol).

Key Concepts

Solving Inequalities

When it comes to solving inequalities, the goal is to isolate the variable on one side of the inequality sign. This task is similar to solving equations, but with one critical difference: the inequality signs. Remember, when you multiply or divide by a negative number, the inequality sign flips direction.

For the inequality \(x-5 \geq 2\), we make use of the addition property of inequality, which states that you can add the same number to both sides of the inequality without changing its truth. Adding 5 to both sides gives us \(x \geq 7\). This statement now tells us that 'x' can be any number greater than or equal to 7. Clear communication of each step here is essential to ensure understanding and the ability to apply the method to similar problems.

Graphing on a Number Line

After finding the solution to an inequality, graphing on a number line presents a visual representation of all possible values that satisfy the inequality. To graph the solution set of \(x \geq 7\), you would:

• Draw a horizontal line to represent the number line.
• Mark the point corresponding to the value 7. Since the inequality includes the number 7 (as indicated by \geq), you would draw a closed circle on 7 to show that it is part of the solution set.
• Draw an arrow starting at the number 7 and extending to the right, signifying all numbers greater than 7 are included in the solution set.

This graphical approach makes it clear and straightforward which numbers fulfill the inequality.

Inequality Symbols

Understanding inequality symbols is crucial when working with inequalities. These symbols determine the relationship between the values on either side of them. The most common symbols are:\

• \(>\) for 'greater than'
• \(<\) for 'less than'
• \(\geq\) for 'greater than or equal to'
• \(\leq\) for 'less than or equal to'

Each symbol serves as a crucial signpost for both solving the inequality and accurately representing its solution set. For example, the \(\geq\) in our original problem \(x-5 \geq 2\) signifies that 'x' is not just greater than 7, but it could also be equal to 7. Clear use and interpretation of these symbols is essential for grasping the full range of possible solutions in an inequality.