Problem 45 - Introductory and Intermediate Algebra for College Students 4th

Question

Use the multiplication property of inequality to solve each inequality and graph the solution set on a number line. $$3 x \geq-21$$

Step-by-Step Solution

Answer

The solution to the inequality is \(x \geq -7\). The graph on a number line would include -7 and extend to the right to indicate all numbers greater than or equal to -7.

Step 1: Divide by 3/ Apply the multiplication property of inequality

As stated in the analysis, the purpose here is to isolate \(x\) on one side. To do that, start by dividing both sides of the inequality by 3. This gives \[x \geq -21 / 3\]

Step 2: Calculate the division

The inverse operation of multiplication is division. So, divide -21 by 3 and this will give the value of x. Simplifying, \[x \geq -7\]

Step 3: Graph the solution on a number line

Now that the solution is found (\(x \geq -7\)), it can be represented on a number line. All the values greater or equal to -7 are the solution.

Key Concepts

Multiplication Property of Inequality

Understanding the multiplication property of inequality is critical when addressing problems involving inequalities. This mathematical principle informs us that multiplying or dividing both sides of an inequality by a positive number will not change the direction of the inequality. Thus, if you have an inequality like 3x \[\geq\] -21, you can divide both sides by 3 to isolate the variable x.

However, it's essential to remember that if you multiply or divide by a negative number, the inequality symbol must be flipped. For example, if we had -3x \[\geq\] -21 and we divided by -3, the \[\geq\] symbol would become \[\leq\] to maintain the inequality's truth. The exercise given 3x \[\geq\] -21 doesn't require flipping the inequality sign because we are dividing by a positive number. By learning to apply this concept effectively, students can tackle a range of inequality problems with confidence.

Isolate the Variable

The goal of solving an inequality is to isolate the variable on one side of the inequality sign. Isolating the variable means manipulating the equation so that the variable you are solving for, typically represented as x, stands alone. In the given exercise, 3x \[\geq\] -21, isolating the variable involves dividing both sides by 3. The division (which is the inverse operation of multiplication) simplifies the inequality to x \[\geq\] -7.

This step is crucial because it not only simplifies the equation but also provides us with a clear statement about the variable: any number greater than or equal to -7 is a solution to the inequality. It's a foundational skill that enables students to solve various algebraic problems beyond just inequalities, making it a cornerstone of algebraic understanding.

Graphing on a Number Line

Once the inequality is simplified, the next step is to graph the solution set on a number line. This visual representation helps in understanding the set of all possible values that satisfy the inequality. For the inequality x \[\geq\] -7, graphing starts by drawing a horizontal line, which represents the number line, and marking the point -7 on it. Because the inequality includes the 'equal to' part (denoted by \[\geq\]), you'll place a closed circle on -7, indicating that -7 is part of the solution.

From -7, you'd draw a line or arrow extending to the right, which signifies all numbers greater than -7. This is because those are the numbers that also satisfy the inequality x \[\geq\] -7. Graphing inequalities on a number line is a powerful way to visualize and understand the range of solutions, and it is a valuable tool for students to grasp the concept of inequalities deeply.