Problem 75 - Introductory and Intermediate Algebra for College Students 4th

Question

Use both the addition and multiplication properties of inequality to solve each inequality and graph the solution set on a number line. $$8 x+3>3(2 x+1)-x+5$$

Step-by-Step Solution

Answer

The solution to the inequality is \(x > \frac{5}{3}\). This can be represented on the number line by marking the number \(5/3\) and shading to the right of it.

Step 1: Distribute and Simplify

Start by distributing in the right side of the inequality. Multiply 3 with every term inside the parenthesis, resulting in \(6x + 3 - x + 5\). Then, combine elements that contain 'x' and constants: \(6x-x+3+5\) to simplify to \(5x+8\). So, the inequality is now: \(8x+3>5x+8\).

Step 2: Rearrange the Inequality

Subtract '5x' and '3' on both sides of the inequality to get all x terms on one side and the constants on the other side. This results in \(8x-5x > 8-3\), which simplifies to \(3x > 5\).

Step 3: Solve for x

To isolate 'x', divide every term by 3. This yields \(x > \frac{5}{3}\). This is the solution to the inequality.

Step 4: Graph the Solution on a Number Line

We can graph this solution on the number line, marking down the number \(5/3\) on the line and shade all values to the right of it since 'x' is greater than \(5/3\). Note, the point at \(5/3\) is not included, so it's an open circle.

Key Concepts

Understanding the Addition Property of Inequality

The Addition Property of Inequality is a simple yet crucial concept in solving inequalities. It states that you can add or subtract the same value from both sides of an inequality without changing the inequality's direction. This principle is useful for simplifying inequalities and isolating variables. For instance, in the original exercise, we used this property when we subtracted the same terms from both sides of the inequality. By subtracting 5x from both sides, we were able to keep the inequality balanced and make it easier to solve.

Mastering the Multiplication Property of Inequality

The Multiplication Property of Inequality is similar to the addition property but involves multiplying or dividing. If you multiply or divide both sides of an inequality by a positive number, the direction of the inequality remains the same. However, always pay attention when multiplying or dividing by a negative number: you must flip the inequality sign. In our example, after simplifying, we used this property when dividing both sides by 3 to solve for x in the expression \(3x > 5\). Since we divided by a positive number, the inequality's direction stayed the same.

The Importance of Graphing Inequalities

Graphing inequalities on a number line provides a visual understanding of the solution set. You start by marking the critical point, which is the solution to the inequality. From there, determine the direction in which to shade, based on the inequality's sign. In this specific exercise, the solution \(x > \frac{5}{3}\) is an open circle at \(\frac{5}{3}\) with shading to the right. This means all numbers greater than \(\frac{5}{3}\) are solutions. Remember, an open circle indicates that the number itself is not included, whereas a closed circle would include it.

Unpacking the Distributive Property

The Distributive Property is utilized when you need to simplify expressions, especially those involving parentheses. It allows you to multiply each term inside a parenthesis by a factor outside, thus removing the parentheses for easier manipulation. For instance, in the given exercise, we distributed the 3 across the terms inside the parentheses \(3(2x+1)\). This resulted in \(6x + 3\), simplifying the expression and aiding the process of solving the inequality.

The Process of Solving Inequalities

Solving inequalities involves several steps, often requiring a combination of different properties such as addition, multiplication, and distribution. First, simplify the inequality by distributing and combining like terms. Next, use the addition or subtraction property to move variables and constants, aiming to isolate the variable on one side. Finally, employ the multiplication property to solve for the variable. In the provided exercise, these steps led us from a complex expression to the simple inequality \(x > \frac{5}{3}\). Each step progressively brought us closer to the solution, showcasing how methodical processes work together to solve inequalities effectively.