Problem 77 - 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. $$\frac{x}{3}-2 \geq 1$$

Step-by-Step Solution

Answer

The solution to the inequality is \(x \geq 9\). The solution on the number line includes the number 9 and all numbers greater than 9.

Step 1: Simplify the Inequality

Begin with the inequality \(\frac{x}{3}-2 \geq 1\). Add 2 to both sides to isolate the term with \(x\) on one side which results in \(\frac{x}{3} \geq 1 + 2\). Simplifying this, we obtain \(\frac{x}{3} \geq 3\).

Step 2: Solve for \(x\)

To solve for \(x\), we multiply both sides of the inequality by 3 to isolate \(x\). Remember that since 3 is positive, the direction of the inequality remains the same. Multiplying both sides by 3, we find \(x \geq 3 \times 3\), or \(x \geq 9\).

Step 3: Graph the Solution

Then we sketch a number line to represent all the possible solutions for \(x\). On the number line, a value of 9 is marked as a filled circle (indicative that 9 is included in the solution), and an arrow is drawn to the right of 9 (indicative that all numbers greater than 9 are also in the solution set).

Key Concepts

Addition Property of Inequality

The addition property of inequality is a powerful tool when solving inequalities. It states that you can add the same number to both sides of an inequality without changing its overall direction. This is similar to solving equations, where you can rearrange terms to isolate variables.
Consider the inequality from the exercise: \( \frac{x}{3} - 2 \geq 1 \). Our first goal is to isolate the term with \( x \), which means we need to move \( -2 \) to the other side. By adding \( 2 \) to both sides, we maintain the inequality's direction and obtain \( \frac{x}{3} \geq 3 \). This shows the real magic of the addition property. It helps in rewriting inequalities in simpler forms without altering the solution set.
Remember, using this property is crucial when you want to move terms without affecting the inequality's integrity. It preserves the order and direction, making it easier for you to solve the inequality.

Multiplication Property of Inequality

When solving inequalities, you might need to multiply or divide both sides by a number to isolate the variable. The multiplication property of inequality allows this, with one important concern: if you multiply or divide by a negative number, you must reverse the inequality sign. In our current example, though, we only multiply by a positive number, so the inequality sign remains unchanged.
After applying the addition property, we have \( \frac{x}{3} \geq 3 \). To isolate \( x \), multiply both sides by 3, the denominator of the fraction. Since 3 is positive, the inequality remains \( x \geq 9 \).
- If multiplying by positive, keep the same direction.- If multiplying by negative, flip the sign.
This property is essential for effectively handling fractions or coefficients in inequalities, ultimately enabling you to express the solution in a straightforward manner.

Graphing Inequalities

Once you solve an inequality, it's important to graph the solution to fully understand the set of possible values. Graphing is a visual representation that helps interpret which numbers satisfy the inequality.
In our exercise, we determined \( x \geq 9 \). When graphing this on a number line:

• Place a filled circle on 9 to indicate that 9 is part of the solution set (because the inequality includes \( \geq \) not just \( > \)).
• Draw a ray or arrow pointing to the right of 9, showing all numbers greater than 9 are included.

Graphing helps to see at a glance the range of valid \( x \) values and to ensure no potential solutions are missed. This practice is incredibly helpful for understanding and communicating solutions to inequalities.