Problem 52 - Introductory Algebra for College Students

Question

Use the multiplication property of inequality to solve each inequality and graph the solution set on a number line. \(-20 x>-140\)

Step-by-Step Solution

Answer

The solution to the inequality is \(x < 7\). This is represented on a number line with an open circle at 7 and an arrow pointing to the left.

Step 1: Divide both sides of the inequality by -20

Isolate x by dividing both sides of the inequality by -20. Ensure you change the direction of the inequality sign because we divided by a negative number. So, \(-20x > -140\) becomes \(x < 7\)

Step 2: Graphing x on a number line

As the inequality is less than 7, so on the number line, you will open circle over 7 (an open circle indicates that 7 is not included in the solution), and you draw an arrow towards the left from the circle. The left direction indicates all numbers less than 7.

Key Concepts

Multiplication Property of Inequality

One of the essential tools in solving inequalities is the multiplication property of inequality. This property states that when you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign.
For example, when we have an inequality like \(-20x > -140\), we multiply or, in this specific case, divide both sides by \(-20\) to solve for \(x\). Remembering to reverse the inequality sign is crucial, so the inequality becomes \(x < 7\).

• Multiply or divide by a positive number: inequality sign remains the same.
• Multiply or divide by a negative number: flip the inequality sign.

Solving Inequalities

Solving an inequality is much like solving an equation, but with a twist—attention to the inequality sign. In the example \(-20x > -140\), our goal is to isolate \(x\) on one side.
We achieved this by dividing both sides by \(-20\), remembering to flip the inequality, resulting in \(x < 7\).
The key steps are:

• Perform algebraic operations as you would for an equation.
• Adjust the direction of the inequality sign when multiplying or dividing by a negative.
• Ensure the solution accurately represents the inequality.

By following these steps, you'll find the solution for the inequality.

Graphing Inequalities

Graphing is a visual way to present the solution of an inequality on a number line. For the inequality \(x < 7\), we need to graph this to convey the solutions clearly.
Use an open circle on the number line at location \(7\). The open circle indicates that \(7\) itself is not part of the solution. Then, extend a line to the left from the circle towards lower values, showing all numbers less than \(7\).

• Open circle for inequalities \(<\) or \(>\), representing that the boundary number is not included.
• Closed circle for \(\leq\) or \(\geq\), indicating inclusion.

This graph provides a quick and intuitive understanding of which numbers satisfy the inequality.

Number Line Representation

Representing inequalities on a number line is a way to visually understand the range of solutions. In our example, \(x < 7\), this is done through a simple diagram:

• Draw a horizontal line to serve as the number line.
• Mark the critical value (in this case, \(7\)).
• Place an open circle at \(7\) to show it is not included.
• Draw an arrow extending leftwards, indicating all values less than \(7\) are solutions.

The number line representation is not just a visual aid; it confirms our understanding about which numbers make the inequality true. It highlights the range of potential values for \(x\) and simplifies the process of communicating our findings.