Problem 67 - 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. \(2 x-5>-x+6\)

Step-by-Step Solution

Answer

The solution to the inequality \(2 x-5>-x+6\) is \(x > \frac{1}{3}\)

Step 1: Add x to both sides

In order to get the x terms on one side, add \(x\) to both sides of inequality. It gives us \(2x-x > -5+6 \) which simplifies to \(3x > 1\).

Step 2: Divide by the coefficient of x

By dividing both sides of the inequality by \(3\) (the coefficient of \(x\)), we can find the value of \(x\). So \(x > \frac{1}{3}\).

Step 3: Graph the Solution

To graph the solution, we draw a number line, mark \(1/3\) on it. Since \(x\) is greater than \(\frac{1}{3}\), we make an open circle at \(\frac{1}{3}\) and shade the region to the right of the open circle.

Key Concepts

Addition Property of Inequality

When working with inequalities, the addition property of inequality is an essential tool. Think of this property as an easy way to maintain the balance of an inequality, similar to how you would handle equations. If you add or subtract the same number from both sides of an inequality, the inequality stays true. In the exercise, we used this property to move the x terms to one side of the inequality.

• Start with the inequality: \(2x - 5 > -x + 6\).
• Add \(x\) (or equivalent expression) to both sides: \(2x + x - 5 > 6\).

This alteration keeps our inequality balanced and simplifies our problem. This step is crucial because it helps isolate the variable, setting the stage for finding the solution.Always remember that this addition or subtraction does not change the inequality sign. Using the addition property can make future steps more straightforward and help you clearly see the inequality's structure.

Multiplication Property of Inequality

The multiplication property of inequality is another fundamental aspect when solving inequalities. This property allows you to multiply or divide both sides of an inequality by the same positive number without changing the inequality's direction. However, be cautious—if you multiply or divide by a negative number, you must flip the inequality sign. In our current example, since we are using a positive number, we do not need to change the inequality sign.In this exercise, after using the addition property, the inequality was simplified to:

• \(3x > 1\)
• Divide by \(3\) to solve for \(x\): \(x > \frac{1}{3}\)

Dividing both sides by the same positive number (in this case, 3) allows us to find the solution without affecting the inequality.Always pay attention to the number you are using; this small detail can significantly impact your results if not careful, particularly when negative numbers are involved.

Graphing Inequalities

Graphing inequalities is a useful way to visually represent the solution of an inequality. It helps you understand what values the solutions can take. In our exercise, after we have determined that \(x > \frac{1}{3}\), we translate this into a graph on a number line.To graph the inequality:

• Draw a number line and mark the point \(\frac{1}{3}\) on it.
• Since \(x\) is greater than (and not equal to) \(\frac{1}{3}\), we use an open circle to indicate that \(\frac{1}{3}\) is not included in the solution set.
• Shade the region to the right of \(\frac{1}{3}\) to show all numbers greater than \(\frac{1}{3}\) are part of the solution set.

This graphical representation of \(x > \frac{1}{3}\) helps make the abstract inequality tangible. It's crucial to remember that an open circle means the value is not included, whereas a closed circle would mean it is included in the solution set. Graphing inequalities makes interpreting solutions more intuitive and is an essential skill for solving complex math problems.