Problem 77 - Introductory Algebra for College Students

Question

Use both the addition and multiplication properties of inequality to solve each inequality and graph the solution set on a number line. \(\frac{x}{3}-2 \geq 1\)

Step-by-Step Solution

Answer

The solution to the inequality is \(x \geq 9\). The graph is a number line with a solid dot at 9 and shaded to the right of 9.

Step 1: Addition property of inequality

Add 2 on both sides of the inequality \(\frac{x}{3} - 2 \geq 1\). This yields: \(\frac{x}{3} \geq 3\)

Step 2: Multiplication property of inequality

Multiply both sides of the inequality \(\frac{x}{3} \geq 3\) by 3. This yields: \(x \geq 9\)

Step 3: Represent the solution on a number line

On a number line, place a solid dot on 9 to signify that 9 is included in the solution set (because of the 'greater than or equal to' symbol) and shade to the right of 9 since that's where numbers larger than 9 are located.

Key Concepts

Understanding the Addition Property of Inequality

Inequalities are expressions that show the relationship between two expressions that aren't necessarily equal. To solve inequalities, we often need to isolate the variable. One of the key methods we use is the addition property of inequality. This property tells us that adding or subtracting the same number from both sides of an inequality keeps the inequality true. For example, in the inequality \[ \frac{x}{3} - 2 \geq 1 \]we can add 2 to both sides to eliminate the '-2' on the left side. This results in:

• \(\frac{x}{3} \geq 3\)

Doing this doesn't change the inequality's direction or meaning; it simply helps us get one step closer to finding the solution.

Applying the Multiplication Property of Inequality

With the variable closer to being isolated, we next use the multiplication property of inequality. This property allows us to multiply or divide both sides of an inequality by the same positive number without changing the direction of the inequality. In our example,

• \(\frac{x}{3} \geq 3\)

we want to eliminate the fraction by multiplying both sides by 3, resulting in:

• \(x \geq 9\)

If you multiply or divide by a negative number, remember that the inequality sign flips direction. Since we're using a positive number here, the sign remains the same, leading us to the solution \(x \geq 9\).

Graphing Inequalities on a Number Line

Graphing inequalities helps visually represent the solution. In the inequality \(x \geq 9\), the symbol "\(\geq\)" means "greater than or equal to." Therefore, 9 is part of our solution. On a number line, we show this by:

• Placing a solid dot on the number 9.
• Shading the line to the right of 9 to express all numbers greater than 9.

The solid dot indicates 9 is included in our solution, unlike an open dot which would show exclusion. Shading right visually demonstrates all possible values of \(x\) that satisfy \(x \geq 9\). This way, you can easily understand and communicate the solution to others.