Problem 51 - 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. $$-16 x>-48$$

Step-by-Step Solution

Answer

The solution to the inequality is \(x < 3\).

Step 1: Division

Divide both sides of the inequality \(-16x > -48\) by -16 to isolate \(x\). Remember, dividing by a negative number changes the direction of the inequality sign. This yields: \(x < 3\)

Step 2: Graphing the inequality on a number line

Draw a number line and mark the point 3. Because the inequality is 'less than' and not 'less than or equal to', make a light circle around the point. Then shade all numbers less than 3 to the left of the point on the number line.

Key Concepts

Multiplication Property of Inequality

The multiplication property of inequality comes into play when solving inequalities that involve multiplication or division. This property states that when you multiply or divide both sides of an inequality by a positive number, the direction of the inequality sign remains the same.
For example, if you have an inequality like \(3x < 9\), you can divide both sides by 3 (a positive number), and the inequality remains \(x < 3\). However, caution is needed when dealing with negative numbers. As we will discuss later, multiplying or dividing by a negative number flips the inequality sign.

Solving Inequalities

To solve inequalities, the goal is similar to solving equations: isolate the variable on one side. This often involves using basic operations like addition, subtraction, multiplication, and division.
Consider the inequality \(-16x > -48\). Here, you need to get \(x\) alone, which involves dividing through by \(-16\) to isolate \(x\).
This brings us to an important rule: when you divide or multiply both sides of an inequality by a negative, the inequality sign flips direction. So for our example, dividing by \(-16\) yields \(x < 3\) instead of \(x > 3\). Keeping this rule in mind is crucial for solving inequalities accurately.

Graphing Inequalities

Graphing an inequality provides a visual representation of the solution. Once you have solved the inequality, you can represent it on a number line for clarity.
For \(x < 3\), draw a number line and mark the number 3. Since the inequality is 'less than' (not 'less than or equal to'), you place an open circle at 3 to indicate that 3 is not included in the solution set. Then shade all numbers to the left of 3, showing all numbers less than 3 are included.
Graphing helps visually affirm the solution set of an inequality and can be a useful tool in understanding number relationships.

Division by Negative Numbers

Dividing by a negative number in an inequality requires special attention. Unlike ordinary division in arithmetic, dividing by a negative in inequalities causes the inequality sign to reverse.
This means if you're solving \(-16x > -48\), the division step flips the greater than sign to a less than sign when dividing by \(-16\), resulting in the inequality \(x < 3\).
Remembering this rule is essential, as it can change the entire solution if not applied correctly. It might help to think of the reversal as compensating for the change in direction, ensuring the inequality relationship stays true.