Problem 29 - 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. $$5 x-9<4 x+7$$

Step-by-Step Solution

Answer

The solution to the inequality is \(x < 16\). This means that all numbers less than 16 are solutions. When graphed on a number line, all values less than 16 are included in the solution set.

Step 1: Regroup terms

First, regroup the inequality so that you have terms containing \(x\) on one side and constants on the other side. Subtract \(4x\) from both sides to get the \(x\) terms together, and add \(9\) to both sides to get the constants together. This gives \[5x - 4x < 7 + 9\].

Step 2: Simplify inequality

Combine like terms on each side to simplify the inequality. This results in \(x < 16\)

Step 3: Draw number line

Draw a number line to represent the solution set. The solution will be all numbers less than 16. Because the inequality is 'less than' but not 'less than or equal to', we will represent 16 with an open circle because it is not included in the solution set. The graph of the solution set is an arrow from the open circle at 16 extending to the left, meaning that all values less than 16 are included.

Key Concepts

Solving Inequalities

When solving inequalities, the main goal is to isolate the variable on one side of the inequality sign. This process is similar to solving equations, but with a few key differences. Here’s a simple way to look at it:

• Just like in equations, perform the same operation on both sides to keep the inequality balanced.
• Use the addition property of inequality, which states you can add or subtract the same number from both sides, without changing the inequality.

Let's look at the example: \(5x - 9 < 4x + 7\).

• Begin by subtracting \(4x\) from both sides. This gets all the \(x\) terms on the left: \(5x - 4x < 7 + 9\).
• Next, simplify by adding \(9\) to both sides, consolidating the constant terms: \(x < 16\).

Once simplified, you'll end up with an inequality like \(x < 16\), highlighting the solutions.

Graphing Inequalities

Graphing inequalities visually represents the solution sets on a number line, making it easier to understand the range of values involved. Here’s how to graph the inequality \(x < 16\):

• First, draw a standard number line.
• Locate the critical number (in this case, 16) on the number line.

For strict inequalities like \(<\) or \(>\), we use open circles to indicate that the number itself isn't part of the solution set. With \(x < 16\), place an open circle at 16, because 16 is not included. Then draw an arrow extending from the open circle to the left, showcasing all numbers less than 16.

Algebra Concepts

Understanding basic algebra concepts when working with inequalities is crucial. These concepts help you perform operations correctly and understand the behavior of inequalities.

• Combine like terms: It simplifies expressions and reveals the inequality in its simplest form.
• Use properties of operations: Such as addition and subtraction, to manipulate and rearrange terms effectively.
• Keep track of inequality direction: Unlike equations, reversing an inequality sign can happen if you multiply or divide by a negative number.

Remember, solving inequalities involves these core algebra concepts that not only make the problem simpler but ensure accuracy throughout the process. Familiarity with these concepts helps you solve and visualize inequalities seamlessly.