Problem 32 - Introductory and Intermediate Algebra for College Students 4th

Question

Use the addition property of inequality to solve each inequality and graph the solution set on a number line. $$8 x-9>7 x-3$$

Step-by-Step Solution

Answer

The solution to the inequality \(8x-9 > 7x-3\) is \(x > 6\). This is represented on the number line as an open circle at 6 with an arrow pointing to the right from 6.

Step 1: Subtraction

Subtract \(7x\) from both sides to isolate all \(x\)-related terms on one side. This gives \(8x - 7x - 9 > -3 \), which simplifies to \(x - 9 > -3\).

Step 2: Addition

Add 9 to both sides to completely isolate \(x\) on one side. This gives \(x > -3 + 9\), which simplifies to \(x > 6\).

Step 3: Graphing

Draw a number line and mark the point 6. Place an open circle at point 6 (because the inequality is 'greater than' and does not include 6). Draw an arrow pointing to the right from 6, indicating all numbers greater than 6 are included in the solution set.

Key Concepts

Addition Property of Inequality

Understanding the Addition Property of Inequality is crucial when it comes to solving inequality problems. The property states that you can add the same number to both sides of an inequality without changing its direction. For instance, if we have an inequality like \( A > B \), and we add the same value, let's call it \( C \), to both sides, we will have \( A + C > B + C \). This principle is essential because it allows for the isolation of variables, I'll discuss in a later section.

This principle was applied in the textbook exercise where we solved the inequality \(8x - 9 > 7x - 3\) by adding 9 to both sides. By doing so, we maintained the balance of the inequality while successfully isolating the variable. With careful manipulation using this property, we can find the correct range of values that satisfy the inequality.

Graphing Solutions on a Number Line

Once we solve the inequality, we illustrate the range of solutions on a number line for a clear visual representation. Graphing is a powerful tool in mathematics as it provides a quick way to understand the relationship and size of different numbers. For inequalities, we use an open or closed circle to show whether the endpoint is included (closed) or not (open) in the solution set.

In our example, after isolating the variable to find \(x > 6\), we graphed this on a number line. The arrow pointing to the right from the open circle at 6 represents all the numbers greater than 6, with the open circle indicating that 6 itself is not part of the solution. This visual aid is incredibly helpful for immediately grasping where the solution set lies, especially for complex inequalities.

Isolation of Variables

The isolation of variables is the key objective when solving equations and inequalities. It involves rearranging the equation so the variable we are solving for is by itself on one side of the equation or inequality. It often involves applying properties of addition, subtraction, multiplication, or division to both sides of the equation.

The original textbook exercise showed us how to isolate the variable 'x' by subtracting \(7x\) from both sides, which simplified our inequality to \(x > 6\). In doing so, it became clear what range of values for 'x' would satisfy the inequality. The goal is always to express the variable in its simplest form without any coefficients or additional terms on the same side, which is precisely what we achieved in the provided steps.