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Inequalities are expressions that show the relationship between two values or expressions that are not equal. Inequalities often involve the symbols less than (<), less than or equal to (≤), greater than (>), or greater than or equal to (≥). In this exercise, we worked with the inequality \( \frac{1}{2}(2x + 3) < 6 \).

To solve such inequalities, our goal is to find the range of values that satisfy the inequality expression. Unlike equations that have a single solution, inequalities typically have a range of solutions, creating a continuous set of possible values. Understanding the representation of inequalities helps interpret this range and potential solutions effectively.

• If the inequality uses a '<' or '>', the solution does not include the boundary value, illustrated by an open circle or point.
• If the inequality uses '≤' or '≥', the boundary value is included, shown with a closed or filled circle.

Being able to represent inequalities correctly is crucial for interpreting and communicating mathematical relationships.

A number line is a visual tool used to represent numbers and their relationships, including inequalities. When solving an inequality, depicting its solutions on a number line can help clarify the range of possible values that satisfy the inequality.

In the solution to our problem, we determined the inequality \( x < 4.5 \). To represent this on a number line:

• First, locate the number 4.5 on the number line.
• Then, use an open circle at 4.5 to show that 4.5 itself is not included in the solution set.
• Shade the region to the left of 4.5 to indicate that all numbers less than 4.5 are included in the solution set.

Using a number line this way visually demonstrates the set of values that make the inequality true, which can be particularly helpful in understanding solutions to complex inequalities.

Algebraic manipulation involves rearranging and simplifying expressions to solve mathematical problems, such as equations and inequalities. In this exercise, it was essential to carefully rearrange terms and solve step-by-step to isolate the variable of interest.

Here's how we manipulated the given inequality:\( \frac{1}{2}(2x + 3) < 6 \):

• First, distribute \( \frac{1}{2} \) to the terms inside the bracket. This changes the expression to \( x + \frac{3}{2} < 6 \).
• Next, isolate \( x \) by subtracting \( \frac{3}{2} \) from both sides to get \( x < 6 - \frac{3}{2} \).
• Simplify the result to find \( x < \frac{9}{2} \) or equivalently \( x < 4.5 \).

Each step in algebraic manipulation requires attention to operational details, ensuring accuracy so that solutions are correct and consistent. Proper mastery of these steps enables you to solve a wide range of mathematical problems involving inequalities.