Problem 49 - 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. \(-3 x \geq 15\)

Step-by-Step Solution

Answer

The solution to the inequality is \( x \leq -5 \). This means that any number less than or equal to -5 will satisfy the inequality.

Step 1: Divide both sides by -3

The first step to solve the inequality \(-3x \geq 15\) is to isolate \( x \) by dividing both sides of the inequality by -3. This yields: \( x \leq -5 \) . Remember, the inequality sign flips when we divide by -3.

Step 2: Graph the solution on a number line

The solution set will be all the numbers less than or equal to -5. On the number line, this is represented by a closed dot on -5 to denote that -5 is included in the solution set, and an arrow extending to the left from this dot.

Key Concepts

Multiplication Property of Inequality

Understanding the multiplication property of inequality is crucial when working with algebraic expressions that involve inequalities. We are often required to isolate the variable on one side to find its possible values. For example, consider the inequality \[\begin{equation}-3x \geq 15.\end{equation}\]To solve for x, you need to get the x alone on one side. In this case, you'd do so by dividing both sides of the inequality by -3. What's critical to note is that when multiplying or dividing both sides of an inequality by a negative number, you must reverse the inequality sign. Thus, dividing by -3, our original inequality \[\begin{equation}-3x \geq 15\end{equation}\] becomes \[\begin{equation}x \leq -5.\end{equation}\]This sign reversal ensures the inequality still holds true after the multiplication or division with a negative number is performed.

The division by a negative is a pivotal step that students sometimes overlook, leading to incorrect solutions. Always remember, the only time you reverse the sign is when you multiply or divide both sides by a negative number. This knowledge will not only help you solve inequalities correctly but also understand why certain solutions are possible.

Number Line Graphing

After solving an inequality, it's helpful to represent the solution visually. Number line graphing is a tool that allows us to do just that. With our previous example, \[\begin{equation}x \leq -5,\end{equation}\]we need to portray all the numbers that are less than or equal to -5. To do this on a number line:

• Place a closed dot on the number -5 to include it in our solution. A closed dot means that the number is part of the solution set.
• Draw an arrow pointing to the left from -5 to show all numbers less than -5 are also included in the solution set.

By doing this, anyone can visually see the range of possible values for x. It is a powerful way to comprehend and communicate solutions, especially when dealing with inequalities. A well-drawn number line simplifies the complex algebraic understanding into a clear visual representation that is easy to grasp and interpret.

Inequality Sign Reversal

The inequality sign reversal is a rule that initially surprises many students. It is a principle stating that when you multiply or divide both sides of an inequality by a negative number, the direction of the inequality sign must be reversed to maintain the truth of the statement. For instance, if you have an inequality like \[\begin{equation}-3x \geq 15,\end{equation}\]don't forget to flip the sign from 'greater than or equal to' to 'less than or equal to' when dividing both sides by -3. This critical step is showcased in the solution: \[\begin{equation}x \leq -5.\end{equation}\]
If you don't flip the sign, your solution will be incorrect, leading to an inaccurate representation of the inequality. This is why paying attention to the sign of the number you're dividing or multiplying by is imperative. Always double-check your work to ensure the inequality sign is properly managed to reflect the correct set of solutions.