Problem 50 - 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. \(-7 x \leq 21\)

Step-by-Step Solution

Answer

The solution of the inequality \(-7x \leq 21\) is \(x \geq -3\) which represents all the numbers greater than or equal to -3.

Step 1: Identify the inequality

The given inequality is \(-7x \leq 21\). We need to isolate x by dividing both sides by -7.

Step 2: Divide by -7

Dividing both sides by -7, we get \(x \geq -3\). Remember, the inequality symbol gets reversed because we divide by a negative number.

Step 3: Graph the solution on a number line

Draw a number line, mark the point -3. Since x is greater than or equal to -3, shade the region on the number line to the right of -3, including -3 because it's a 'greater than or equal' inequality.

Key Concepts

Multiplication Property of Inequality

When solving inequalities, the multiplication property of inequality is a fundamental concept that allows for the manipulation of inequalities in a similar fashion to equations. In essence, this property states that you can multiply or divide both sides of an inequality by the same positive number without changing the direction of the inequality sign.

For example, if you have an inequality like \(5x < 20\), you can divide both sides by 5 to isolate \(x\), yielding \(x < 4\). The inequality sign remains the same direction because the number 5 is positive.

This property is incredibly useful when you need to solve an inequality involving a variable. However, it is crucial to remember that this property only applies directly when the number you are multiplying or dividing by is positive. When the number is negative, there is an additional rule to consider, which is the subject of our next section.

Dividing by a Negative Number

Dividing by a negative number when working with inequalities introduces a twist to the multiplication property of inequality. Whenever you divide (or multiply) both sides of an inequality by a negative number, the direction of the inequality sign must be reversed.

This is a crucial step that is often overlooked, so pay close attention. If you consider the inequality from our exercise, \(-7x \leq 21\), when we divide both sides by -7, what we are really doing is applying an operation that changes the relational aspect of the inequality. Therefore, the inequality becomes \(x \geq -3\), as you would reverse the \(\leq\) sign into a \(\geq\) sign.

This reversal is necessary because multiplying or dividing by a negative number effectively 'flips' the number line, so to speak, making larger numbers smaller and vice versa. Always keeping this in mind is essential for solving inequalities correctly.

Number Line Graphing

Number line graphing is a visual method to represent the solution set of an inequality. A number line helps to easily understand and display the set of all possible values that satisfy the inequality.

When graphing \(x \geq -3\), as per our example, you would start by drawing a horizontal line, which is the number line. You'd then mark the point corresponding to -3. Since the inequality includes 'greater than or equal to,' you would draw a solid dot at -3 to indicate that -3 is included in the set of solutions.

After marking the point, you would shade the region of the number line that extends to the right of -3, demonstrating that all numbers greater than -3 are also part of the solution set. The direction of the shading aligns with the direction of the inequality sign. Graphing the solutions on a number line not only provides a clear answer but also reinforces the concept of the inequality's solution range.