Problem 51 - 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. \(-16 x>-4\)

Step-by-Step Solution

Answer

The solution to the inequality -16x > -4 is \(x < 0.25\). This is represented graphically on the number line with an open circle at 0.25, and all numbers to the left of this point.

Step 1: Understand the Problem

We're given the inequality -16x > -4. We're asked to solve this inequality and graph its solution on a number line.

Step 2: Apply the Multiplication Property of Inequality

The multiplication property of inequality states that when you multiply both sides of an inequality by a negative number, the direction of the inequality sign is reversed. So, let's divide both sides of our inequality by negative 16. While doing so, remember to reverse the inequality sign. \[\frac{-16x}{-16} < \frac{-4}{-16}\]. This simplifies to \[x < 0.25\].

Step 3: Plot the Solution on a Number Line

Now, let's graph the solution on a number line. Draw a number line and mark 0.25 on it. Because x is strictly less than 0.25 and not equal to 0.25, we represent this on the number line with an open circle at 0.25. Then draw an arrow to the left to represent all the numbers less than 0.25.

Key Concepts

Multiplication Property of Inequality

The multiplication property of inequality is a foundational concept when solving algebraic inequalities. It tells us that when we multiply or divide both sides of an inequality by the same positive number, the direction of the inequality remains unchanged. However, when the number is negative, the inequality reverses direction. For example, if we have an inequality like \( -2x > 6 \), and we divide both sides by -2 (a negative number), we must flip the inequality sign to get \( x < -3 \). This property is essential to ensure that the inequality still holds true after the operation.

When solving an inequality like \( -16x > -4 \), and we divide both sides by -16, according to the multiplication property of inequality, we must reverse the inequality to \( x < 0.25 \). This step is critical in maintaining the correct relationships between the numbers involved and leading us toward the right solution.

Number Line Graphing

Number line graphing is a visual tool that aids in understanding the range of solutions to an inequality. Graphing on a number line helps students to easily see which numbers satisfy the inequality. When graphing \( x < 0.25 \) on a number line, an open circle is placed at 0.25 to indicate that 0.25 is not part of the solution set. Then, a line or arrow is drawn to the left of the open circle, implying that all numbers to the left of 0.25 are solutions to the inequality. This graphic representation is intuitive as it physically demonstrates the 'less than' relationship by showing all numbers that are indeed less than 0.25.

Inequality Reversal

Inequality reversal is a crucial concept when dealing with inequalities in algebra. As seen in the problem \( -16x > -4 \), when both sides of an inequality are multiplied or divided by a negative number, the inequality sign must be flipped. This rule prevents logical inconsistencies that could arise from negatives affecting the balance of the inequality. It's essential to remember this rule as failing to reverse the inequality sign will result in incorrect solutions. A helpful tip for students is to note the reversal immediately when dividing or multiplying by a negative to avoid missing this crucial step.

Algebraic Problem Solving

Algebraic problem solving is a step-by-step approach to find the solution to mathematical statements involving unknowns. It encompasses understanding the problem, identifying what is being asked, determining the steps required, and systematically carrying out those steps to arrive at the solution. In the context of inequalities, it involves applying rules such as the multiplication property of inequality and graphing solutions on a number line.

Working through the problem \( -16x > -4 \) involves recognizing the need for inequality reversal upon dividing by a negative, simplifying to find \( x < 0.25 \), and then accurately representing this on a number line. Comprehending and applying these methods develops strong problem-solving skills that are central to algebra and critical for mathematical success in more advanced topics.